Skip to main content
SpringerLink
Log in

Additive Number Theory

  • Chapter
Unsolved Problems in Number Theory

Part of the book series: Problem Books in Mathematics ((1605,volume 1))

Abstract

One of the most infamous problems is Goldbach’s conjecture that every even number greater than 4 is expressible as the sum of two odd primes. Richstein has verified it up to 4 · 1014, and on 2003-10-03 I learnt that Oliviera e Silva has extended this to 6 · 1016. Vinogradov proved that every odd number greater than \({3^{{3^{15}}}}\) is the sum of three primes and Chen Jing-Run has shown that all large enough even numbers are the sum of a prime and the product of at most two primes. Wang & Chen have reduced the number \({3^{{3^{15}}}}\) to e 114 ≈ 3.23274 × 1049 under the assumption of the generalized Riemann hypothesis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 99.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • R. C. Baker & Glyn Harman, The three primes theorem with almost equal summands, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356(1998) 763–780; MR 99c: 1 1124.

    Google Scholar 

  • R. C. Baker, Glyn Harman & J. Pintz, The exceptional set for Goldbach’s problem in short intervals, Sieve methods, exponential sums and their applications in number theory (Cardiff, 1995) 87–100, London Math. Soc. Lecture Note Ser., 237, Cambridge Univ. Press, 1997; MR 99g:11121.

    Google Scholar 

  • Claus Bauer, On Goldbach’s conjecture in arithmetic progressions, Studia Math. Sci. Hungar., 37(2001) 1–20; MR 2002c:11124.

    Google Scholar 

  • J. Bohman & C.-E. Froberg, Numerical results on the Goldbach conjecture, BIT, 15(1975) 239–243.

    MathSciNet  MATH  Google Scholar 

  • Cai Ying-Chun & Lu Ming-Gao, On Chen’s theorem, Analytic number theory (Beijing/Kyoto, 1999), 99–119, Dev. Math., 6(2002), Kluwer, Dordrecht; MR 2003a:11130.

    Google Scholar 

  • Chen Jing-Run, On the representation of a large even number as the sum of a prime and the product of at most two primes, Sci. Sinica, 16(1973) 157–176; MR 55 #7959; II, 21(1978) 421–430; MR 80e:10037.

    Google Scholar 

  • Chen Jing-Run & Wang Tian-Ze, On the Goldbach problem (Chinese), Acta Math. Sinica, 32(1989) 702–718; MR 91e:11108.

    Google Scholar 

  • Chen Jing-Run & Wang Tian-Ze, The Goldbach problem for odd numbers (Chinese), Acta Math. Sinica, 39(1996) 169–174; MR 97k:11138.

    Google Scholar 

  • J. G. van der Corput, Sur l’hypothèse de Goldbach pour presque tous les nombres pairs, Acta Arith., 2(1937) 266–290.

    Google Scholar 

  • N. G. Cudakov, On the density of the set of even numbers which are not representable as the sum of two odd primes, Izv. Akad. Nauk SSSR Ser. Mat.,2(1938) 25–40.

    Google Scholar 

  • N. G. Cudakov, On Goldbach-Vinogradov’s theorem, Ann. Math.(2), 48 (1947) 515–545; MR9, 11.

    Google Scholar 

  • Jean-Marc Deshouillers, G. Effinger, H. T. T. te Riele & D. Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electronic Research Announcements of the AMS (www.ams.org/era), page 99

    Google Scholar 

  • Jean-Marc Deshouillers, Andrew Granville, Wladyslaw Narkiewicz & Carl Pomerance, An upper bound in Goldbach’s problem, Math. Comput., 61(1993) 209–213; MR 94b:11101.

    Google Scholar 

  • Gunter Dufner & Dieter Wolke, Einige Bemerkungen zum Goldbachschen Problem, Monatsh. Math., 118(1994) 75–82; MR 95h:11106.

    Google Scholar 

  • T. Estermann, On Goldbach’s problem: proof that almost all even positive integers are sums of two primes, Proc. London Math. Soc.(2),44(1938) 307–314.

    MathSciNet  Google Scholar 

  • S. Yu. Fatkina, On a generalization of Goldbach’s ternary problem for almost equal summands, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 2001 no.2, 22–28, 70; transl. Moscow Univ. Math. Bull., 56(2001) 22–29; MR 2002c:11127.

    Google Scholar 

  • Antonio Filz, Problem 1046, J. Recreational Math., 14(1982) 64; 15(1983) 71.

    Google Scholar 

  • D. A. Goldston, Linnik’s theorem on Goldbach numbers in short intervals, Glasgow Math. J., 32(1990) 285–297; MR 911:11134.

    Google Scholar 

  • Dan A. Goldston, On Hardy and Littlewood’s contribution to the Goldbach conjecture, Proc. Amalfi Conf. Anal. Number Theory (Maiori 1989), 115–155, Univ. Salerno, 1992; MR 94m:11122.

    Google Scholar 

  • A. Granville, J. van de Lune & H. J. J. te Riele, Checking the Goldbach conjecture on a vector computer, in R. A. Mollin (ed.), Number Theory and Applications, NATO ASI Series, Kluwer, Boston, 1989, pp. 423–433; MR 93c:11085.

    Google Scholar 

  • Richard K. Guy, Prime Pyramids, Crux Mathematicorum, 19(1993) 97–99.

    Google Scholar 

  • D. R. Heath-Brown & J.-C. Puchta, Integers represented as a sum of primes and powers of two, Asian J. Math., 6(2002) 535–565; MR 2003i:11146.

    Google Scholar 

  • Christopher Hooley, On an almost pure sieve, Acta Arith., 66(1994) 359–368; MR 95g:11092.

    Google Scholar 

  • Jia Chao-Hua, Goldbach numbers in short interval, I, Sci. China Ser. A, 38(1995) 385–406; II, 513–523; MR 96f:11130, 11131.

    Google Scholar 

  • Jia Chao-Hua, On the exceptional set of Goldbach numbers in a short interval, Acta Arith., 77(1996) 207–287; MR 97e:11126.

    Google Scholar 

  • J. Kaczorowski, A. Perelli & J. Pintz, A note on the exceptional set for Goldbach’s problem in short intervals, Monatsh. Math., 116(1993) 275–282; MR 95c:11136; corrigendum, 119(1995) 215–216.

    Google Scholar 

  • Leszek Kaniecki, On Snirelman’s constant under the Riemann hypothesis, Acta Arith., 72(1995) 361–374; MR 961:11112.

    Google Scholar 

  • Margaret J. Kenney, Student Math Notes, NCTM News Bulletin, Nov. 1986.

    Google Scholar 

  • Li Hong-Ze, Goldbach numbers in short intervals, Sci. China Ser. A, 38(1995) 641–652; MR 96h:11100.

    Google Scholar 

  • Liu Jian-Ya & Liu Ming-Chit, Representation of even integers as sums of squares of primes and powers of 2, J. Number Theory, 83(2000) 202–225; MR 2001g:11156.

    Google Scholar 

  • Liu Jian-Ya & Liu Ming-Chit, Representation of even integers by cubes of primes and powers of 2, Acta Math. Hungar., 91(2001) 217–243; MR 2003c:11131.

    Google Scholar 

  • Liu Ming-Chit & Wang Tian-Ze, On the Vinogradov bound in the three primes Goldbach conjecture, Acta Arith., 105(2002) 133–175; MR 20031:11147.

    Google Scholar 

  • Hiroshi Mikawa, On the exceptional set in Goldbach’s problem, Tsukuba J. Math., 16(1992) 513–543; MR 94c:11098.

    Google Scholar 

  • Hiroshi Mikawa, Sums of two primes (Japanese), Si rikaisekikenkyûsho Kôkyûroku No. 886(1994) 149–169; MR 96g:11123.

    Google Scholar 

  • H. L. Montgomery & R. C. Vaughan, The exceptional set in Goldbach’s problem, Acta Arith., 27(1975) 353–370.

    MathSciNet  MATH  Google Scholar 

  • Pan Cheng-Dong, Ding Xia-Xi & Wang Yuan, On the representation of every large even integer as the sum of a prime and an almost prime, Sci. Sinica, 18(1975) 599–610; MR 57 #5897.

    Google Scholar 

  • A. Perelli & J. Pintz, On the exceptional set for Goldbach’s problem in short intervals, J. London Math. Soc., 47(1993) 41–49; MR 93m:11104.

    Google Scholar 

  • J. Pintz, & I. Z. Ruzsa, On Linnik’s approximation to Goldbach’s problem, I, Acta Arith., 109(2003) 169–194; MR 2004c:11185.

    Google Scholar 

  • Olivier Ramaré, On Snirel’man’s constant Ann. Scuola Norm. Sup. Pisa Cl. Sci.(2), 22(1995) 645–706; MR 97a:11167.

    Google Scholar 

  • Jörg Richstein, Verifying the Goldbach conjecture up to 4. 1014, Math. Comput., 70(2001) 1745–1749 (electronic); MR 2002c:11131.

    Google Scholar 

  • Jörg Richstein, Computing the number of Goldbach partitions up to 5 • 108, Algorithmic Number Theory (Leiden, 2000), Springer Lecture Notes in Comput. Sci., 1838(2000) 475–490; MR 2002g:11142.

    Google Scholar 

  • P. M. Ross, On Chen’s theorem that every large even number has the form p 1 + p 2 or p 1 + p 2p3, J. London Math. Soc., (2) 10(1975) 500–506.

    MathSciNet  MATH  Google Scholar 

  • Yannick Saouter, Checking the odd Goldbach conjecture up to 1020, Math. Comput., 67(1998) 863–866; MR 98g:11115.

    Google Scholar 

  • Shan Zun & Kan Jia-Hai, On the representation of a large even number as the sum of a prime and an almost prime: the prime belongs to a fixed arithmetic progression, Acta Math. Sinica, 40(1997) 625–638.

    Google Scholar 

  • Matti K. Sinisalo, Checking the Goldbach conjecture up to 4 • 1011, Math. Comput., 61(1993) 931–934.

    MathSciNet  MATH  Google Scholar 

  • Mels Sluyser, Parity of prime numbers and Goldbach’s conjecture, Math. Sci., 19(1994) 64–65; MR 95f:11004.

    Google Scholar 

  • M. L. Stein & P. R. Stein, New experimental results on the Goldbach conjecture, Math. Mag., 38(1965) 72–80; MR 32 #4109.

    Google Scholar 

  • M. L. Stein & P. R. Stein, Experimental results on additive 2-bases, Math. Comput., 19(1965) 427–434.

    MATH  Google Scholar 

  • D. I. Tolev, On the number of representations of an odd integer as the sum of three primes, one of which belongs to an arithmetic progression, Tr. Mat. Inst. Steklova 218(1997) Anal. Teor. Chisel i Prilozh., 415–432.

    Google Scholar 

  • Robert C. Vaughan, On Goldbach’s problem, Acta Arith., 22(1972) 21–48.

    MathSciNet  Google Scholar 

  • Robert C. Vaughan, A new estimate for the exceptional set in Goldbach’s problem, Proc. Symp. Pure Math.,(Analytic Number Theory, St. Louis), Amer. Math. Soc., 24(1972) 315–319.

    Google Scholar 

  • I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR,15(1937) 169–172.

    MATH  Google Scholar 

  • I. M. Vinogradov, Some theorems concerning the theory of primes, Mat. Sb. N.S., 2(44) (1937) 179–195.

    MATH  Google Scholar 

  • Morris Wald, Problem 1664, J. Recreational Math., 20(1987–88) 227–228; Solution 21(1988–89) 236–237.

    Google Scholar 

  • Edward T. H. Wang, Advanced Problem 6189, Amer. Math. Monthly 85 (1978) 54 [no solution has appeared].

    Google Scholar 

  • Wang Tian-Ze, The odd Goldbach problem for odd numbers under the generalized Riemann hypothesis, II, Adv. in Math., 25(1996) 339–346; MR 98h:11125.

    Google Scholar 

  • Wang Tian-Ze & Chen Jing-Run, An odd Goldbach problem under the general Riemann hypothesis, Sci. China Ser. A, 36(1993) 682–691; MR 95a:11090.

    Google Scholar 

  • Yan Song-Y., A simple verification method for the Goldbach conjecture, Internat. J. Math. Ed. Sci., 25(1994) 681–688; MR 95e:11109.

    Google Scholar 

  • Yu Xin-He, A new approach to Goldbach’s conjecture, Fujian Shifan Daxue Xuebao Ziran Keine Ban, 9(1993) 1–8; MR 95e:11110.

    Google Scholar 

  • Zhang Zhen-Feng & Wang Tian-Ze, The ternary Goldbach problem with primes in arithmetic progressions, Acta Math. Sin. (Engl. Ser.), 17(2001) 679–696.

    MathSciNet  MATH  Google Scholar 

  • Dan Zwillinger, A Goldbach conjecture using twin primes, Math. Comput., 33(1979) 1071; MR 80b:10071.

    Google Scholar 

  • W. E. Briggs, Prime-like sequences generated by a sieve process, Duke Math. J., 30(1963) 297–312; MR 26 #6145.

    Google Scholar 

  • R. G. Buschman & M. C. Wunderlich, Sieve-generated sequences with translated intervals, Canad. J. Math., 19(1967) 559–570; MR 35 #2855.

    Google Scholar 

  • R. G. Buschman & M. C. Wunderlich, Sieves with generalized intervals, Boll. Un. Mat. Ital.(3), 21(1966) 362–367.

    Google Scholar 

  • Paul Erdös & Eri Jabotinsky, On sequences of integers generated by a sieving process, I, II, Nederl. Akad. Wetensch. Proc. Ser. A, 61 = Indag. Math., 20(1958) 115–128; MR 21 #2628.

    Google Scholar 

  • Verna Gardiner, R. Lazarus, N. Metropolis & S. Ulam, On certain sequences of integers generated by sieves, Math. Mag., 31(1956) 117–122; 17,711.

    Google Scholar 

  • David Hawkins & W. E. Briggs, The lucky number theorem, Math. Mag., 31(1957–58) 81–84, 277–280; MR 21 #2629, 2630.

    Google Scholar 

  • D. Hawkins, The random sieve, Math. Mag., 31(1958) 1–3.

    MathSciNet  Google Scholar 

  • C. C. Heyde, Ann. Probability, 6(1978) 850–875.

    MathSciNet  Google Scholar 

  • M. C. Wunderlich, Sieve generated sequences, Canad. J. Math., 18(1966) 291–299; MR 32 #5625.

    Google Scholar 

  • M. C. Wunderlich, A general class of sieve-generated sequences, Acta Arith., 16(1969–70) 41–56; MR 39 #6852.

    Google Scholar 

  • M. C. Wunderlich & W. E. Briggs, Second and third term approximations of sieve-generated sequences, Illinois J. Math., 10(1966) 694–700; MR 34 #153.

    Google Scholar 

  • Julien Cassaigne & Steven R. Finch, A class of 1-additive sequences and quadratic recurrences, Experiment. Math., 4(1995) 49–60; MR 96g:11007.

    Google Scholar 

  • Steven R. Finch, Conjectures about s-additive sequences, Fibonacci Quart., 29(1991) 209–214; MR 92j:11009.

    Google Scholar 

  • Steven R. Finch, Are 0-additive sequences always regular? Amer. Math. Monthly, 99 (1992) 671–673.

    Google Scholar 

  • Steven R. Finch, On the regularity of certain 1-additive sequences, J. Combin. Theory Ser. A, 60(1992) 123–130; MR 93c:11009.

    Google Scholar 

  • Steven R. Finch, Patterns in 1-additive sequences, Experiment. Math., 1(1992) 57–63; MR 93h11014.

    Google Scholar 

  • P. N. Muller, On properties of unique summation sequences, M.Sc. thesis, University of Buffalo, 1966.

    Google Scholar 

  • Frank W. Owens, Sums of consecutive U-numbers, Abstract 74T-A74, Notices Amer. Math. Soc., 21(1974) A-298.

    Google Scholar 

  • Raymond Queneau, Sur les suites s-additives, J. Combin. Theory, 12(1972) 31–71; MR 46 #1741.

    Google Scholar 

  • Bernardo Recam6,n, Questions on a sequence of Ulam, Amer. Math. Monthly, 80(1973) 919–920.

    Google Scholar 

  • James Schmerl & Eugene Spiegel, The regularity of some 1-additive sequences, J. Combin. Theory Ser. A, 66(1994) 172–175; MR 95h:11010.

    Google Scholar 

  • S. M. Ulam, Problems in Modern Mathematics, Interscience, New York, 1964, p. ix.

    Google Scholar 

  • Marvin C. Wunderlich, The improbable behaviour of Ulam’s summation sequence, in Computers and Number Theory, Academic Press, 1971, 249–257.

    Google Scholar 

  • David Zeitlin, Ulam’s sequence {U n }, U 1 = 1, U 2 = 2, is a complete sequence, Abstract 75T-A267, Notices, Amer. Math. Soc., 22(1975) A-707.

    Google Scholar 

  • I. N. Baker, Solutions of the functional equation (f (x))2 — f (x 2) = h(x), Canad. Math. Bull., 3(1960) 113–120.

    Google Scholar 

  • E. Bolker, The finite Radon transform, Contemp. Math., 63(1987) 27–50; MR 88b:51009.

    Google Scholar 

  • J. Boman, E. Bolker & P. O’Neil, The combinatorial Radon transform modulo the symmetric group, Adv. Appl. Math., 12(1991) 400–411; MR 92m:05014.

    Google Scholar 

  • Jan Boman & Svante Linusson, Examples of non-uniqueness for the combinatorial Radon transform modulo the symmetric group, Math. Scand., 78(1996) 207–212; MR 97g:05014.

    Google Scholar 

  • John A. Ewell, On the determination of sets by sets of sums of fixed order, Canad. J. Math., 20(1968) 596–611.

    Google Scholar 

  • B. Gordon, A. S. Fraenkel & E. G. Straus, On the determination of sets by the sets of sums of a certain order, Pacific J. Math., 12(1962) 187–196; MR 27 #3576.

    Google Scholar 

  • Ross A. Honsberger, A gem from combinatorics, Bull. ICA, 1(1991) 56–58.

    Google Scholar 

  • J. Lambek & L. Moser, On some two way classifications of the integers, Canad. Math. Bull., 2(1959) 85–89.

    MathSciNet  MATH  Google Scholar 

  • B. Liu & X. Zhang, On harmonious labelings of graphs, Ars Combin., 36(1993) 315–326.

    MathSciNet  MATH  Google Scholar 

  • L. Moser, Problem E1248, Amer. Math. Monthly, 64(1957) 507.

    Google Scholar 

  • J. Ossowski, On a problem of Galvin, Congressus Numerantium, 96(1993) 65–74.

    MathSciNet  MATH  Google Scholar 

  • D. G. Rogers, A functional equation: solution to Problem 89–19*, SIAM Review, 32(1990) 684–686.

    Google Scholar 

  • J. L. Selfridge & E. G. Straus, On the determination of numbers by their sums of a fixed order, Pacific J. Math., 8(1958) 847–856; MR 22 #4657.

    Google Scholar 

  • Walter Aiello & M. V. Subbarao, A conjecture in addition chains related to Scholz’s conjecture, Math. Comput., 61(1993) 17–23; MR 93k:11015.

    Google Scholar 

  • Hatem M. Bahig, Mohamed H. El-Zahar & Ken Nakamula, Some results for some conjectures in addition chains, Combinatorics, computability and logic (Constanta, 2001) 47–54, Springer Ser. Discrete Math. Theor. Comput. Sci., 2001; MR 2003i:11031.

    Google Scholar 

  • F. Bergeron & J. Berstel, Efficient computation of addition chains, J. Théor. Nombres Bordeaux, 6(1994) 21–38; MR 95m:11144.

    Google Scholar 

  • A. T. Brauer, On addition chains, Bull. Amer. Math. Soc., 45(1939) 736–739.

    MathSciNet  Google Scholar 

  • A. Cottrell, A lower bound for the Scholz-Brauer problem, Abstract 73T-A200, Notices Amer. Math. Soc., 20(1973) A-476.

    Google Scholar 

  • Paul Erdts, Remarks on number theory III. On addition chains, Acta Arith., 6(1960) 77–81.

    MathSciNet  Google Scholar 

  • R. P. Giese, PhD thesis, University of Houston, 1972.

    Google Scholar 

  • R. P. Giese, A sequence of counterexamples of Knuth’s modification of Utz’s conjecture, Abstract 72T-A257, Notices Amer. Math. Soc., 19(1972) A-688.

    Google Scholar 

  • A. A. Gioia & M. V. Subbarao, The Scholz-Brauer problem in addition chains II, Congr. Numer. XXII, Proc. 8th Manitoba Conf. Numer. Math. Comput. 1978, 251–274; MR 80i: 10078; Zbl. 408:10037.

    Google Scholar 

  • A. A. Gioia, M. V. Subbarao & M. Sugunamma, The Scholz-Brauer problem in addition chains, Duke Math. J., 29(1962) 481–487; MR 25 #3898.

    Google Scholar 

  • W. Hansen, Zum Scholz-Brauerschen Problem, J. reine angew. Math., 202(1959) 129–136; MR 25 #2027.

    Google Scholar 

  • Kevin Hebb, Some problems on addition chains, thesis, Univ. of Alberta, 1974.

    Google Scholar 

  • A. M. Ilin, On additive number chains (Russian), Problemy Kibernet., 13 (1965) 245–248.

    Google Scholar 

  • H. Kato, On addition chains, PhD dissertation, Univ. Southern California, June 1970.

    Google Scholar 

  • Donald Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading MA, 1969, 398–422.

    MATH  Google Scholar 

  • D. J. Newman, Computing when multiplications cost nothing, Math. Comput., 46(1986) 255–257.

    MATH  Google Scholar 

  • Arnold Scholz, Aufgabe 253, Jber. Deutsch. Math.-Verein. II, 47(1937) 41–42 (supplement).

    Google Scholar 

  • Rolf Sonntag, Theorie der Addition Sketten, PhD Technische Universität Hannover, 1975.

    Google Scholar 

  • K. B. Stolarsky, A lower bound for the Scholz-Brauer problem, Canad. J. Math., 21(1969) 675–683; MR 40 #114.

    Google Scholar 

  • M. V. Subbarao, Addition chains — some results and problems, in R. A. Mollin (ed.) Number Theory and Applications, NATO ASI Series, Kluwer, Boston, 1989, pp. 555–574; MR 93a:11105.

    Google Scholar 

  • E. G. Straus, Addition chains of vectors, Amer. Math. Monthly 71(1964) 806–808.

    MathSciNet  Google Scholar 

  • E. G. Thurber, The Scholz-Brauer problem on addition chains, Pacific J. Math., 49(1973) 229–242; MR 49 #7233.

    Google Scholar 

  • E. G. Thurber, On addition chains l(mn) < l(n) − b and lower bounds for c(r), Duke Math. J., 40(1973) 907–913.

    Google Scholar 

  • E. G. Thurber, Addition chains and solutions of l(2n) = l(n) and l(2n − 1) = n + l(n) − 1, Discrete Math., 16(1976) 279–289; MR 55 #5570; Zbl. 346.10032.

    Google Scholar 

  • W. R. Utz, A note on the Scholz-Brauer problem in addition chains, Proc. Amer. Math. Soc., 4(1953) 462–463; MR 14, 949.

    Google Scholar 

  • C. T. Wyburn, A note on addition chains, Proc. Amer. Math. Soc., 16(1965) 1134.

    MathSciNet  Google Scholar 

  • Matthias Beck, Ricardo Diaz & Sinai Robins, The Frobenius problem, rational polytopes, and Fourier-Dedekind sums, J. Number Theory, 96(2002) 1–21; MR 2003j:11117.

    Google Scholar 

  • Matthias Beck, Ira M. Gessel & Takao Komatsu, The polynomial part of a restricted partition function related to the Frobenius problem, Electron. J. Combin., 8(2001) N7, 5 pp.; MR 2002f:05017.

    Google Scholar 

  • E. R. Berlekamp, J. H. Conway & R. K. Guy, The Emperor and his money, chap.18 of Winning Ways for your Mathematical Plays, 2nd ed., AK Peters, Natick MA, 2003.

    Google Scholar 

  • Alfred Brauer, On a problem of partitions, Amer. J. Math., 64(1942) 299312.

    Google Scholar 

  • A. Brauer & J. E. Shockley, On a problem of Frobenius, J. reine angew. Math., 211(1962) 215–220; MR 26 #6113.

    Google Scholar 

  • T. C. Brown & Peter Shiue Jau-Shyong, A remark related to the Frobenius problem, Fibonacci Quart., 31(1993) 32–36; MR 93k:11018.

    Google Scholar 

  • Chen Qing-Hua & Liu Yu-Ji, A result on the Frobenius problem, Fujian Shifan Daxue Xuebao Ziran Kerwe Ban, 11(1995) 33–37; MR 97k:11016.

    Google Scholar 

  • Frank Curtis, On formulas for the Frobenius number of a numerical semi-group, Math. Scand., 67(1990) 190–192; MR 92e:11019.

    Google Scholar 

  • J. L. Davison, On the linear Diophantine problem of Frobenius, J. Number Theory, 48(1994) 353–363; MR 95j:11033.

    Google Scholar 

  • J. Dixmier, Proof of a conjecture of Erdös and Graham concerning the problem of Frobenius, J. Number Theory, 34(1990) 198–209; MR 91g:11007.

    Google Scholar 

  • P. Erdös & R. L. Graham, On a linear diophantine problem of Frobenius, Acta Arith., 21(1972) 399–408; MR 47 #127.

    Google Scholar 

  • Harold Greenberg, Solution to a linear diophantine equation for nonnegative integers, J. Algorithms, 9(1988) 343–353; MR 89j:11122.

    Google Scholar 

  • B. R. Heap & M. S. Lynn, On a linear Diophantine problem of Frobenius: An improved algorithm, Numer. Math., 7(1965) 226–231; MR 31 #1227.

    Google Scholar 

  • Mihâly Hujter & Béla Vizâri, The exact solutions to the Frobenius problem with three variables, J. Ramanujan Math. Soc., 2(1987) 117–143; MR 89f:11039.

    Google Scholar 

  • S. M. Johnson, A linear diophantine problem, Canad. J. Math., 12(1960) 390–398; MR 22 #12074.

    Google Scholar 

  • I. D. Kan, The Frobenius problem for classes of polynomial solvability, Mat. Zametki, 70(2001) 845–853; Math. Notes, 70(2001) 771–778; MR 2002k:11034.

    Google Scholar 

  • I. D. Kan, The Frobenius problem for classes of polynomial solvability. (Russian) Mat. Zametki, 70(2001) 845–853; transi. Math. Notes, 70(2001) 771–778; MR 2002k:11034.

    Google Scholar 

  • I. D. Kan, Boris S. Stechkin & I. V. Sharkov, On the Frobenius problem for three arguments, Mat. Zametki, 62(1997) 626–629; transl. Math. Notes, 62(1997) 521–523; MR 99h:11022.

    Google Scholar 

  • Ravi Kannan, The Frobenius problem, Foundations of software technology and theoretical computer science (Bangalore, 1989) 242–251, Lecture Notes in Comput. Sci., 405 Springer, Berlin, 1989; MR 91e:68063.

    Google Scholar 

  • Ravi Kannan, Lattice translates of a polytope and the Frobenius problem, Combinatorica, 12(1992) 161–177; MR 93k:52015.

    Google Scholar 

  • Géza Kiss, On the extremal Frobenius problem in a new aspect, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 45(2002) 139–142 (2003).

    Google Scholar 

  • Vsevolod F. Lev, On the extremal aspect of the Frobenius problem, J. Corn-bin. Theory Ser. A, 73(1996) 111–119; MR 97d:11045.

    Google Scholar 

  • Vsevolod F. Lev, Structure theorem for multiple addition and the Frobenius problem, J. Number Theory, 58(1996) 79–88; addendum 65(1997) 96–100; MR 97d:11056, 98e:11045.

    Google Scholar 

  • Vsevolod F. Lev & P. Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith., 70(1995) 85–91.

    MathSciNet  MATH  Google Scholar 

  • Mordechai Lewin, On a linear diophantine problem, Bull. London Math. Soc., 5(1973) 75–78; MR 47 #3311.

    Google Scholar 

  • Mordechai Lewin, An algorithm for a solution of a problem of Frobenius, J. reine angew. Math., 276(1975) 68–82; MR 52 #259.

    Google Scholar 

  • Albert Nijenhuis, A minimal-path algorithm for the “money changing problem”, Amer. Math. Monthly, 86(1979) 832–835; correction, 87(1980) 377; MR 82m:68070ab.

    Google Scholar 

  • Niu Xue-Feng, A formula for the largest integer which cannot be represented as \(\sum\nolimits_{i = 1}^s {{a_i}{x_i}} ,\) Heilongjiang Daxue Ziran Kexue Xuebao, 9(1992) 28–32; MR 93k:11019.

    Google Scholar 

  • Marek Raczunas & Piotr Chrwtowski-Wachtel, A Diophantine problem of Frobenius in terms of the least common multiple, Discrete Math., 150(1996) 347–357; MR 97d:11057.

    Google Scholar 

  • J. L. Ramirez-Alfonsfn, Complexity of the Frobenius problem, Combinatorica, 16(1996) 143–147; MR 97d:11058.

    Google Scholar 

  • Stefan Matthias Ritter, On a linear Diophantine problem of Frobenius: extending the basis, J. Number Theory, 69(1998) 201–212; MR 99b:11023.

    Google Scholar 

  • J. B. Roberts, Note on linear forms, Proc. Amer. Math. Soc., 7(1956) 465–469.

    MathSciNet  MATH  Google Scholar 

  • O. J. Rodseth, On a linear Diophantine problem of Frobenius, J. reine angew. Math., 301(1978) 171–178; MR 58 #27741.

    Google Scholar 

  • Oystein J. Rodseth, A note on T. C. Brown & P. J.-S. Shiue’s paper: “A remark related to the Frobenius problem”, Fibonacci Quart., 32(1994) 407–408; MR 95j:11022.

    Google Scholar 

  • Ernst S. Selmer, On the linear diophantine problem of Frobenius, J. reine angew. Math., 293/294(1977) 1–17; MR 56 #246.

    Google Scholar 

  • E. S. Selmer & O. Beyer, On the linear diophantine problem of Frobenius in three variables, J. reine angew. Math., 301(1978) 161–170.

    MathSciNet  MATH  Google Scholar 

  • J. J. Sylvester, Math. Quest. Educ. Times, 37(1884) 26; (not 41(1884) 21 as in Dickson and elsewhere).

    Google Scholar 

  • Calogero Tinaglia, Some results on a linear problem of Frobenius, Geometry Seminars, 1996–1997, Univ. Stud. Bologna, 1998, 231–244; MR 99e:11026.

    Google Scholar 

  • Amitabha Tripathi, On a variation of the coin exchange problem for arithmetic progressions, Integers, 3(2003) Al; MR 2003k:11018.

    Google Scholar 

  • M. D. Atkinson, A. Negro & N. Santoro, Sums of lexicographically ordered sets, Discrete Math., 80(1990) 115–122.

    MathSciNet  MATH  Google Scholar 

  • Jaegug Bae, On subset-sum-distinct sequences, Analytic Number Theory, Vol. 1 (Allerton Park, 1995) 31–37, Progr. Math., 138, Birkhäuser Boston MA, 1996; MR 97d:11016.

    Google Scholar 

  • Jaegug Bae, An extremal problem for subset-sum-distinct sequences with congruence conditions, Discrete Math., 189(1998) 1–20; MR 99h:11018.

    Google Scholar 

  • Jaegug Bae, A compactness result for a set of subset-sum-distinct sequences, Bull. Korean Math. Soc., 35(1998) 515–525; MR 99k:11018.

    Google Scholar 

  • Jaegug Bae, On maximal subset-sum-distinct sequences, Commun. Korean Math. Soc., 17(2002) 215–220; MR 2003b:11014.

    Google Scholar 

  • Jaegug Bae, On generalized subset-sum-distinct sequences, Int. J. Pure Appl. Math., 1(2002) 343–352; MR 2003c:11022.

    Google Scholar 

  • Tom Bohman, A sum packing problem of Erdös and the Conway-Guy sequence, Proc. Amer. Math. Soc., 124(1996) 3627–3636; MR 97b:11027.

    Google Scholar 

  • Thomas A. Bohman, A construction for sets of integers with distinct subset sums, Electronic J. Combin., 5(1998) no. 1, Res. Paper 3, 14pp.

    Google Scholar 

  • Peter Borwein & Michael J. Mossinghoff, Newman polynomials with prescribed vanishing and integer sets with distinct subset sums, Math. Comput., 72(2003) 787–800.

    MathSciNet  MATH  Google Scholar 

  • Chen Yong-Gao, Distinct subset sums and an inequality for convex functions, Proc. Amer. Math. Soc., 128(2000) 2897–2898; MR 2000m:11023.

    Google Scholar 

  • J. H. Conway & R. K. Guy, Sets of natural numbers with distinct sums, Notices Amer. Math. Soc., 15(1968) 345.

    Google Scholar 

  • J. H. Conway & R. K. Guy, Solution of a problem of P. Erdös, Colloq. Math., 20(1969) 307.

    Google Scholar 

  • P. Erdös, Problems and results in additive number theory, Colloq. Théorie des Nombres, Bruxelles, 1955, Liège & Paris, 1956, 127–137, esp. p. 137.

    Google Scholar 

  • Noam Elkies, An improved lower bound on the greatest element of a sum-distinct set of fixed order, J. Combin. Theory Ser. A, 41(1986) 89–94; MR 87b:05012.

    MathSciNet  MATH  Google Scholar 

  • P. E. Frenkel, Integer sets with distinct subset sums, Proc. Amer. Math. Soc., 126(1998) 3199–3200.

    MathSciNet  MATH  Google Scholar 

  • Martin Gardner, Number 5, Science Fiction Puzzle Tales, Penguin, 1981.

    Google Scholar 

  • Hansraj Gupta, Some sequences with distinct sums, Indian J. Pure Math., 5(1974) 1093–1109; MR 57 #12440.

    Google Scholar 

  • Richard K. Guy, Sets of integers whose subsets have distinct sums, Ann. Discrete Math. 12(1982) 141–154

    MathSciNet  MATH  Google Scholar 

  • Y. O. Hamidoune, The representation of some integers as a subset sum, Bull. London Math. Soc., 26(1994) 557–563.

    MathSciNet  MATH  Google Scholar 

  • B. Lindström, On a combinatorial problem in number theory, Canad. Math. Bull., 8(1965) 477–490.

    MathSciNet  MATH  Google Scholar 

  • B. Lindström, Om et problem av Erdös for talfoljder, Nordisk. Mat. Tidskrift, 161–2(1968) 29–30, 80.

    Google Scholar 

  • W. Fred Lunnon, Integer sets with distinct subset-sums, Math. Comput., 50(1988) 297–320.

    MathSciNet  MATH  Google Scholar 

  • Roy Maltby, Bigger and better subset-sum-distinct sets, Mathematika, 44(1997) 56–60; MR 98i:11012.

    MathSciNet  MATH  Google Scholar 

  • Melvyn B. Nathanson, Inverse theorems for subset sums, Trans. Amer. Math. Soc., 347(1995) 1409–1418.

    MathSciNet  MATH  Google Scholar 

  • Paul Smith, Problem E 2536*, Amer. Math. Monthly 82(1975) 300. Solutions and comments, 83(1976) 484.

    Google Scholar 

  • Joshua Von Korff, Classification of greedy subset-sum-distinct-sequences, Discrete Math., 271(2003) 271–282.

    Google Scholar 

  • Harvey L. Abbott, Sidon sets, Canad. Math. Bull., 33(1990) 335–341; MR 91k:11022.

    Google Scholar 

  • Miklos Ajtai, Janos Komlôs & Endre Szemerédi, A dense infinite Sidon sequence, European J. Combin., 2(1981) 1–11; MR 83f:10056.

    Google Scholar 

  • Noga Alon, Independent sets in regular graphs and sum-free subsets of finite groups, Israel J. Math., 73(1991) 247–256; MR 92k:11024.

    Google Scholar 

  • R. C. Bose & S. Chowla, Theorems in the additive theory of numbers, Comment. Math. Helv., 37(1962–63) 141–147.

    Google Scholar 

  • John R. Burke, A remark on B2-sequences in GF[p, x], Acta Arith., 70(1995) 93–96; MR 96f:11028.

    MathSciNet  MATH  Google Scholar 

  • Neil J. Calkin & Jan McDonald Thomson, Counting generalized sum-free sets, J. Number Theory, 68(1998) 151–159; MR 98m:11010.

    MathSciNet  MATH  Google Scholar 

  • Sheng Chen, A note on B2k-sequences, J. Number Theory, 56(1996) 1–3; MR 97a:11035.

    MathSciNet  MATH  Google Scholar 

  • Fan Chung, Paul Erdös & Ronald Graham, On sparse sets hitting linear form, Number Theory for the Millenium, I (Urbana IL, 2000) 257–272, AK Peters, Natick MA, 2002; MR 2003k:11012.

    Google Scholar 

  • Javier Cilleruelo, B2-sequences whose terms are squares, Acta Arith., 55 (1990) 261–265; MR 911:11023.

    Google Scholar 

  • J. Cilleruelo, B2[g] sequences whose terms are squares, Acta Math. Hungar., 67(1995) 79–83; MR 95m:11032.

    Google Scholar 

  • Javier Cilleruelo, New upper bounds for finite Bh, sequences, Adv. Math., 159(2001) 1–17; MR 2002g:11023.

    Google Scholar 

  • Javier Cilleruelo & Antonio Cordoba, B2 [oc]-sequences of square numbers Acta Arith., 61(1992) 265–270; MR 93g:11014.

    Google Scholar 

  • Ben Green, The Cameron-Erdös conjecture, Bull. London Math. Soc., (to appear).

    Google Scholar 

  • Javier Cilleruelo & Jorge Jiménez-Urroz, Bh [g]-sequences, Mathematika, 47(2000) 109–115; MR 2003k:11014.

    Google Scholar 

  • Javier Cilleruelo, Imre Z. Ruzsa & Carlos Trujillo, Upper and lower bounds for finite Bh[g] sequences, J. Number Theory 97(2002) 26–34; MR 2003i:11033.

    Google Scholar 

  • Javier Cilleruelo & Carlos Trujillo, Infinite B2[g] sequences, Israel J. Math., 126(2001) 263–267; MR 2003d:11032.

    Google Scholar 

  • Martin Dowd, Questions related to the Erdös-Turân conjecture, SIAM J. Discrete Math., 1(1988) 142–150; MR 89h:11006.

    Google Scholar 

  • P. Erdös, Some of my forgotten problems in number theory, Hardy Ramanujan J., 15 (1992) 34–50.

    MathSciNet  MATH  Google Scholar 

  • P. Erdös & R. Freud, On sums of a Sidon-sequence, J. Number Theory, 38(1991) 196–205.

    MathSciNet  MATH  Google Scholar 

  • P. Erdös & R. Freud, On Sidon-sequences and related problems (Hungarian), Mat. Lapok, 2(1991) 1–44.

    Google Scholar 

  • P. Erdös & W. H. J. Fuchs, On a problem of additive number theory, J. London Math. Soc., 31(1956) 67–73; MR 17, 586d.

    Google Scholar 

  • Paul Erdös, Melvyn B. Nathanson & Prasad Tetali, Independence of solution sets and minimal asymptotic bases, Acta Arith., 69(1995) 243–258; MR 96e:11014.

    Google Scholar 

  • Paul Erdös, A. Sârközy & V. T. Sôs, On sum sets of Sidon sets, I, J. Number Theory, 47(1994) 329–347; II, Israel J. Math., 90(1995) 221–233; MR 96f:11034.

    Google Scholar 

  • P. Erdös & E. Szemerédi, The number of solutions of \(m = \sum\nolimits_{i = 1}^k {x_i^k} ,\) Proc. Symp. Pure Math. Amer. Math. Soc., 24(1973) 83–90; MR 49 #2529.

    Google Scholar 

  • Paul Erdös & Prasad Tetali, Representations of integers as the sum of k terms, Random Structures Algorithms, 1(1990) 245–261; MR 92c:11012.

    Google Scholar 

  • P. Erdös & P. Turân, On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc., 16(1941) 212–215; MR 3, 270. Addendum, 19(1944) 208; MR 7, 242.

    Google Scholar 

  • H. Furstenberg, From the Erdös-Turân conjecture to ergodic theory the contribution of combinatorial number theory to dynamics, Paul Erdös and his mathematics I (Budapest, 1999), 261–277, Bolyai Soc. Math. Stud., 11, Janos Bolyai Math. Soc., Budapest, 2002; MR 2003j:37010.

    Google Scholar 

  • S. Gouda & Mahmoud Abdel-Hamid Amer, A theorem on the h-range of Bh-sequences, Ann. Univ. Sci. Budapest. Sect. Comput., 15(1995) 65–69; MR 97k:11026.

    Google Scholar 

  • R. L. Graham & N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Discrete Math., 1(1980) 382–404.

    Google Scholar 

  • S.W. Graham, Bh-sequences, Analytic Number Theory, Vol. 1 (Allerton Park, 1995) 431–439, Progr. Math., 138, Birkhäuser Boston MA, 1996; MR 97h:11019.

    Google Scholar 

  • Ben Green, The number of squares and Bh[g] sets, Acta Arith., 100(2001) 365–390; MR 2003d:11033.

    Google Scholar 

  • G. Grekos, L. Haddad, C. Helou & J. Pihko, On the Erdös-Turân conjecture, J. Number Theory, 102(2003) 339–352.

    MathSciNet  MATH  Google Scholar 

  • Laurent Habsieger & Alain Plagne, Ensembles B2 [2]: l’étau se reserre, Integers, 2(2002) Paper A2, 20pp.; MR 2002m:11010.

    Google Scholar 

  • H. Halberstam & K. F. Roth, Sequences, 2nd Edition, Springer, New York, 1982, Chapter 2.

    Google Scholar 

  • Elmer K. Hayashi, Omega theorems for the iterated additive convolution of a nonnegative arithmetic function, J. Number Theory, 13(1981) 176–191; MR 82k:10067.

    Google Scholar 

  • Martin Helm, Some remarks on the Erdös-Turân conjecture, Acta Arith., 63 (1993) 373–378; MR 94c:11012.

    Google Scholar 

  • Gabor Horvath, On a theorem of Erdös and Fuchs, Acta Arith., 103(2002) 321–328; MR 2003k:11019.

    Google Scholar 

  • Gabor Horvath, On a generalization of a theorem of Erdös and Fuchs, Acta Math. Hungar., 92(2001) 83–110; MR 2003m:11017.

    Google Scholar 

  • Jia Xing-De, Some problems and results on subsets of asymptotic bases, Qufu Shifan Daxue Xuebao Ziran Kexue Ban, 13(1987) 45–49; MR 88k:11015.

    Google Scholar 

  • Jia Xing-De, On the distribution of a B2-sequence, Qufu Shifan Daxue Xuebao Ziran Kexue Ban, 14(1988) 12–18; MR 89j:11023.

    Google Scholar 

  • Jia Xing-De, On finite Sidon sequences, J. Number Thoery, 44(1993) 84–92.

    Google Scholar 

  • F. Krückeberg, B2-Folgen und verwandte Zahlenfolgen, J. reine angew. Math., 206(1961) 53–60.

    Google Scholar 

  • Vsevolod Lev & Tomasz Schoen, Cameron-Erdös modulo a prime, Finite Fields Appl., 8(2002) 108–119; MR 2002k:11029.

    Google Scholar 

  • B. Lindström, An inequality for B2-sequences, J. Combin. Theory, 6(1969) 211–212; MR 38 #4436.

    Google Scholar 

  • Bernt Lindström, Bh[g]-sequences from Bh-sequences. Proc. Amer. Math. Soc., 128(2000) 657–659; MR 2000e:11022.

    Google Scholar 

  • Bernt Lindström, Well distribution of Sidon sets in residue classes. J. Number Theory, 69(1998) 197–200; MR 99c:11021.

    Google Scholar 

  • Bernt Lindström, Finding finite B2-sequences faster, Math. Comput., 67(1998) 1173–1178; MR 98m:11012.

    Google Scholar 

  • Greg Martin & Kevin O’Bryant, Continuous Ramsey theory and Sidon sets, 70-page preprint, 2002.

    Google Scholar 

  • M. B. Nathanson, Unique representation bases for the integers, Acta Arith., 108(2003) 1–8; MR 2004c:11013.

    Google Scholar 

  • K. G. Omel/yanov & A. A. Sapozhenko, On the number of sum-free sets in an interval of natural numbers, Diskret. Mat., 14(2002) 3–7; translation in Discrete Math. Appl., 12(2002) 319–323; MR 2003m:11015.

    Google Scholar 

  • Imre Z. Ruzsa, A just basis, Monatsh. Math., 109(1990) 145–151; MR 91e:11016.

    Google Scholar 

  • Imre Z. Ruzsa, Solving a linear equation in a set of integers I, Acta Arith., 65(1993) 259–282; MR 94k:11112.

    Google Scholar 

  • Imre Z. Ruzsa, Sumsets of Sidon sets, Acta Arith., 77(1996)353–359; MR 97j:11013.

    Google Scholar 

  • Imre Z. Ruzsa, An infinite Sidon sequence, J. Number Theory, 68(1998) 6371; MR 99a:11014.

    Google Scholar 

  • Imre Z. Ruzsa, A small maximal Sidon set, Ramanujan J., 2(1998) 55–58; MR 99g:11026.

    Google Scholar 

  • A. Sârközy, On a theorem of Erdös and Fuchs, Acta Arith., 37(1980) 333–338; MR 82a:10066.

    Google Scholar 

  • A. Sârközy, Finite addition theorems, II, J. Number Theory, 48(1994) 197218; MR 95k:11010.

    Google Scholar 

  • Tomasz Schoen, The distribution of dense Sidon subsets of Zm, Arch. Math. (Basel), 79(2002) 171–174; MR 2003f:11022.

    Google Scholar 

  • J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc., 43(1938) 377–385; Zbl 19, 5.

    Google Scholar 

  • Vera T. Sôs, An additive problem in different structures, Graph Theory, Cornbinatorics, Algorithms, and Applications, (SIAM Conf., San Francisco, 1989), 1991, 486–510; MR 92k:11026.

    Google Scholar 

  • Zhang Zhen-Xiang, Finding finite B2-sequences with larger m — a1m2, Math. Comput., 63(1994) 403–414; MR 94i:11109.

    Google Scholar 

  • L. D. Baumert, Cyclic Difference Sets, Springer Lect. Notes Math. 182, New York, 1971.

    MATH  Google Scholar 

  • M. R. Best, A. E. Brouwer, F. J. MacWilliams, A. M. Odlyzko & N. J. A. Sloane, Bounds for binary codes of length less than 25, IEEE Trans. Inform. Theory, IT-24(1978) 81–93.

    MathSciNet  MATH  Google Scholar 

  • F. T. Boesch & Li Xiao-Ming, On the length of Golomb’s rulers, Math. Appl., 2(1989) 57–61; MR 91h:11015.

    Google Scholar 

  • J. H. Conway & N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, New York, 1988.

    MATH  Google Scholar 

  • P. Erdös & H. Hanani, On a limit theorem in combinatorical analysis, Publ. Math. Debrecen, 10(1963) 10–13.

    MathSciNet  Google Scholar 

  • T. A. Evans & H. Mann, On simple difference sets, Sankhya, 11(1951) 357–364; MR 13, 899.

    Google Scholar 

  • Richard A. Gibbs & Peter J. Slater, Distinct distance sets in a graph, Discrete Math., 93(1991) 155–165.

    MathSciNet  MATH  Google Scholar 

  • M. J. E. Golay, Note on the representation of 1, 2,..., n by differences, J. London Math. Soc.(2), 4(1972) 729–734; MR 45 #6784.

    Google Scholar 

  • R. L. Graham & N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, IT-26(1980) 37–43; MR 81d:94026.

    Google Scholar 

  • R. L. Graham & N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J. Algebraic Discrete Methods, 1(1982) 382–404; MR 82f:10067.

    Google Scholar 

  • M. Hall, Cyclic projective planes, Duke Math. J., 14(1947) 1079–1090; MR 9, 370.

    Google Scholar 

  • S. V. Konyagin & A. P. Savin, Universal rulers (Russian), Mat. Zametki, 45(1989) 136–137; MR 90k:51038.

    Google Scholar 

  • John Leech, On the representation of 1, 2,..., n by differences, J. London Math. Soc., 31(1956) 160–169; MR 19, 942f.

    Google Scholar 

  • John Leech, Another tree labelling problem, Amer. Math. Monthly 82(1975) 923–925.

    Google Scholar 

  • F. J. MacWilliams & N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977.

    Google Scholar 

  • J. C. P. Miller, Difference bases: three problems in the additive theory of numbers, A. O. L. Atkin & B. J. Birch (editors) Computers in Number Theory, Academic Press, London, 1971, pp. 299–322; MR 47 #4817.

    Google Scholar 

  • J. Schönheim, On maximal systems of k-tuples, Stud. Sci. Math. Hungar., 1(1966) 363–368.

    MATH  Google Scholar 

  • R. G. Stanton, J. G. Kalbfleisch & R. C. Mullin, Covering and packing designs, Proc. 2nd Conf. Combin. Math. Appl., Chapel Hill, 1970, 428–450.

    Google Scholar 

  • Herbert Taylor, A distinct distance set of 9 nodes in a tree of diameter 36, Discrete Math., 93(1991) 167–168.

    MathSciNet  MATH  Google Scholar 

  • B. Wichmann, A note on restricted difference bases, J. London Math. Soc., 38(1963) 465–466; MR 28 #2080.

    Google Scholar 

  • W. C. Babcock, Intermodulation interference in radio systems, Bell Systems Tech. J., 32(1953) 63–73.

    Google Scholar 

  • R. C. Bose & S. Chowla, Report Inst. Theory of Numbers, Univ. of Colorado, Boulder, 1959, p. 335.

    Google Scholar 

  • R. C. Bose & S. Chowla, Theorems in the additive theory of numbers, Comment. Math. Helvet., 37(1962–63) 141–147.

    MathSciNet  MATH  Google Scholar 

  • Sheng Chen, On size of finite Sidon sequences, Proc. Amer. Math. Soc., (to appear).

    Google Scholar 

  • Sheng Chen, A note on B2k-sequences, J. Number Theory, bf(1994)

    Google Scholar 

  • Sheng Chen, On Sidon sequences of even orders, Acta Arith., 64(1993) 325–330.

    MathSciNet  MATH  Google Scholar 

  • Chen Wen-De Si Torliev Klove, Lower bounds on multiple difference sets, Discrete Math., 98(1991) 9–21; MR 93a:5028.

    Google Scholar 

  • S. W. Graham, Upper bounds for B3–sequences, Abstract 882–11–25, Abstracts Amer. Math. Soc., 14(1993) 416.

    Google Scholar 

  • S. W. Graham, Upper bounds for Sidon sequences, (preprint 1993).

    Google Scholar 

  • D. Hajela, Some remarks on Bh[g]-sequences, J. Number Theory, 29(1988) 311–323; MR 90d:11022.

    Google Scholar 

  • Martin Helm, On B2k-sequences, Acta Arith., 63(1993) 367–371; MR 95c:11029.

    Google Scholar 

  • Martin Helm, A remark on B2k-sequences, J. Number Theory, 49(1994) 246–249; MR 96b:11024.

    Google Scholar 

  • Martin Helm, On B3-sequences, Analytic Number Theory, Vol. 2 (Allerton Park, 1995) 465–469, Progr. Math., 139, Birkhäuer, Boston, 1996; MR 97d:11040.

    Google Scholar 

  • Martin Helm, On the distribution of B3-sequences, J. Number Theory, 58 (1996) 124–129; MR 97d:1041.

    Google Scholar 

  • Jia Xing-De, On B6-sequences, Qufu Shifan Daxue Xuebao Ziran Kexue Ban, 15(1989) 7–11; MR 90j:11022.

    Google Scholar 

  • Jia Xing-De, on B2k-sequences, J. Number Theory, (to appear).

    Google Scholar 

  • T. Klove, Constructions of Bh[g]-sequences, Acta Arith., 58(1991) 65–78; MR 92f:11033.

    Google Scholar 

  • Li An-Ping, On B3-sequences, Acta Math. Sinica, 34(1991) 67–71; MR 92f: 11037.

    Google Scholar 

  • B. Lindström, A remark on B4-sequences, J. Combin. Theory, 7(1969) 276–277.

    Google Scholar 

  • John C. M. Nash, On B4-sequences, Canad. Math. Bull., 32(1989) 446–449; MR 91e: 11025.

    Google Scholar 

  • Jukka Pihko, Proof of a conjecture of Selmer, Acta Arith., 65(1993) 117–135; MR 94k:11010.

    Google Scholar 

  • Alain Plagne, Removing one element from an exact additive basis, J. Number Theory, 87(2001) 306–314; MR 2002d:11014.

    Google Scholar 

  • Imre Z. Ruzsa, An infinite Sidon sequence, J. Number Theory, 68(1998) 6371; MR 99a:11014.

    Google Scholar 

  • I. Z. Ruzsa, An almost polynomial Sidon sequence, Studia Sci. Math. Hungar., 38(2001) 367–375; MR 2002k:11030.

    Google Scholar 

  • Richard Warlimont, Uber eine die Basen der natürlich Zahlen betreffende Ungleichung von Alfred Stöhr, Arch. Math. (Basel) 62(1994) 227–229; MR 94m: 11017.

    Google Scholar 

  • R. Windecker, Eine Abschnittsbasis dritter Ordnung, Kongel. Norske Vidensk. Selsk. Skrifter, 9(1976) 1–3.

    Google Scholar 

  • M. F. Challis, Two new techniques for computing extremal h-bases Ak, Comput. J., 36(1993) 117–126. [An updating of the appendix is available from the author.]

    MATH  Google Scholar 

  • M. Djawadi & G. Hofmeister, The postage stamp problem, Mainzer Seminarberichte, Additive Zahlentheorie, 3(1993) 187–195.

    MATH  Google Scholar 

  • P. Erdös & R. L. Graham, On bases with an exact order, Acta Arith., 37(1980) 201–207; MR 82e:10093.

    Google Scholar 

  • P. Erdös & M. B. Nathanson, Partitions of bases into disjoint unions of bases, J. Number Theory, 29(1988) 1–9; MR 89f:11025.

    Google Scholar 

  • Paul Erdös & Melvyn B. Nathanson, Additive bases with many representations, Acta Arith., 52(1989) 399–406; MR 91e:11015.

    Google Scholar 

  • Georges Grekos, Extremal problems about additive bases, Acta Math. Inform. Univ. Ostraviensis, 6(1998) 87–92; MR 2002d:11013.

    Google Scholar 

  • N. Hämmerer & G. Hofmeister, Zu einer Vermutung von Rohrbach, J. reine angew. Math., 286/287(1976) 239–247; MR 54 #10181.

    Google Scholar 

  • G. Hofmeister, Asymptotische Abschätzungen für dreielementige extremalbasen in natürlichen Zahlen, J. reine angew. Math., 232(1968) 77–101; MR 38 #1068.

    Google Scholar 

  • G. Hofmeister, Die dreielementigen Extremalbasen, J. reine angew. Math., 339(1983) 207–214.

    MathSciNet  MATH  Google Scholar 

  • G. Hofmeister, C. Kirfel & H. Kolsdorf, Extremale Reichweitenbasen, No. 60, Dept. Pure Math., Univ. Bergen, 1991.

    Google Scholar 

  • Jia Xing-De, On a combinatorial problem of Erdös and Nathanson, Chinese Ann. Math. Ser. A, 9(1988) 555–560; MR 90i:11018.

    Google Scholar 

  • Jia Xing-De, On the order of subsets of asymptotic bases, J. Number Theory 37(1991) 37–46; MR 92d:11006.

    Google Scholar 

  • Jia Xing-De, Some remarks on minimal bases and maximal nonbases of integers, Discrete Math., 122(1993) 357–362; MR 94k:11013.

    Google Scholar 

  • Jia Xing-De & Melvyn B. Nathanson, A simple construction of minimal asymptotic bases, Acta Arith., 52(1989) 95–101; MR 90g:11020.

    Google Scholar 

  • Jia Xing-De & Melvyn B. Nathanson, Addition theorems for a-finite groups, The Rademacher legacy to mathematics (University Park, 1992) Contemp. Math., 166, Amer. Math. Soc., 1994, 275–284.

    Google Scholar 

  • Christoph Kirfel, Extremale asymptotische Reichweitenbasen, Acta Arith., 60(1992) 279–288; MR 92m:11012.

    Google Scholar 

  • W. Klotz, Extremalbasen mit fester Elementeanzahl, J. reine andew. Math., 237(1969) 194–220.

    MathSciNet  MATH  Google Scholar 

  • W. Klotz, Eine obere Schranke für die Reichweite einer Extremalbasis zweiter Ordnung, J. reine angew. Math., 238(1969) 161–168; MR 40 #117, 116.

    Google Scholar 

  • W. F. Lunnon, A postage stamp problem, Comput. J., 12(1969) 377–380; MR 40 #6745.

    Google Scholar 

  • L. Moser, On the representation of 1, 2,..., n by sums, Acta Arith., 6(1960) 11–13; MR 23 #A133.

    Google Scholar 

  • Robert W. Owens, An algorithm to solve the Frobenius problem, Math. Mag., 76(2003) 264–275.

    Google Scholar 

  • L. Moser, J. R. Pounder & J. Riddell, On the cardinality of h-bases for n, J. London Math. Soc., 44(1969) 397–407; MR 39 #162.

    Google Scholar 

  • S. Mossige, Algorithms for computing the h-range of the postage stamp problem, Math. Comput., 36(1981) 575–582; MR 82e:10095.

    Google Scholar 

  • S. Mossige, On extremal h-bases A4, Math. Scand., 61(1987) 5–16; MR 89e:11008.

    Google Scholar 

  • Svein Mossige, The postage stamp problem: the extremal basis for k = 4, J. Number Theory, 90(2001) 44–61; MR 2003c:11009.

    Google Scholar 

  • A. Mrose, Untere Schranken für die Reichweiten von Extremalbasen fester Ordnung, Abh. Math. Sem. Univ. Hamburg, 48(1979) 118–124; MR 80g:10058.

    Google Scholar 

  • J. C. M. Nash, Some applications of a theorem of M. Kneser, J. Number Theory, 44(1993) 1–8; MR 94k:11015.

    Google Scholar 

  • John C. M. Nash, Freiman’s theorem answers a question of Erdös, J. Number Theory, 27(1987) 7–8; MR 88k:11014.

    Google Scholar 

  • Melvyn B. Nathanson, Extremal properties for bases in additive number theory, Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. Janos Bolyai 51(1990) 437–446; MR 91h:11009.

    Google Scholar 

  • H. O. Pollak, The postage stamp problem, Consortium, 69(1999) 3–5.

    Google Scholar 

  • J. Riddell & C. Chan, Some extremal 2-bases, Math. Comput., 32(1978) 630–634; MR 57 #16244.

    Google Scholar 

  • O. Rodseth, On h-bases for n, Math. Scand., 48(1981) 165–183; MR 82m: 10034.

    Google Scholar 

  • Oystein J. Rodseth, An upper bound for the h-range of the postage stamp problem, Acta Arith. 54(1990) 301–306; MR 91h:11013.

    Google Scholar 

  • H. Rohrbach, Ein Beitrag zur additiven Zahlentheorie, Math. Z., 42(1937) 1-30; Zbl. 15, 200.

    Google Scholar 

  • H. Rohrbach, Anwendung eines Satzes der additiven Zahlentheorie auf eine gruppentheoretische Frage, Math. Z., 42(1937) 538–542; Zbl. 16, 156.

    Google Scholar 

  • Ernst S. Selmer, On the postage stamp problem with three stamp denominations, Math. Scand., 47(1980) 29–71; MR 82d:10046.

    Google Scholar 

  • Ernst S. Selmer, The Local Postage Stamp Problem, Part I General Theory, Part II The Bases A3 and A4, Part III Supplementary Volume, No. 42, 44, 57, Dept. Pure Math., Univ. Bergen, (86–04–15, 86–09–15, 90–06–12) ISSN 0332–5047. (Available on request.)

    Google Scholar 

  • Ernst S. Selmer, On Stöhr’s recurrent h-bases for N, Kongel. Norske Vidensk. Selsk., Skr. 3, 1986, 15 pp.

    Google Scholar 

  • Ernst S. Selmer, Associate bases in the postage stamp problem, J. Number Theory, 42(1992) 320–336; MR 94b:11012.

    Google Scholar 

  • Ernst S. Selmer & Arne Rödne, On the postage stamp problem with three stamp denominations, II, Math. Scand., 53(1983) 145–156; MR 85j:11075.

    Google Scholar 

  • Ernst S. Selmer & Björg Kristin Selvik, On Rödseth’s h-bases Ak {1, a2, 2a2,..., (k — 2)a2, ak}, Math. Scand., 68(1991) 180–186; MR 92k:11010.

    Google Scholar 

  • Jeffrey Shallit, The computational complexity of the local postage stamp problem, SIGACT News, 33(2002) 90–94.

    Google Scholar 

  • H. S. Shulz, The postage-stamp problem, number theory, and the programmable calculator, Math. Teacher, 92(1999) 20–26.

    Google Scholar 

  • Alfred Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe I, II, J. reine angew. Math., 194(1955) 40–65, 111–140; MR 17, 713.

    Google Scholar 

  • Joseph A. Gallian, A survey — recent results, conjectures and open problems in labeling graphs, J. Graph Theory, 13(1989) 491–504; MR 90k:05132.

    Google Scholar 

  • B. Liu & X. Zhang, On harmonious labelings of graphs, Ars Combin., 36(1993) 315–326.

    MathSciNet  MATH  Google Scholar 

  • Harvey L. Abbott, On a conjecture of Funar concerning generalized sum-free sets, Niew Arch. Wisk.(4), 4(1991) 249–252.

    MathSciNet  Google Scholar 

  • Noga Alon & Daniel J. Kleitman, Sum-free subsets, A Tribute to Paul Erdös, 13–26, Cambridge Univ. Press, 1990; MR 92f:11020.

    Google Scholar 

  • Jean Bourgain, Estimates related to sumfree subsets of sets of integers, Israel J. Math., 97(1997) 71–92; MR 97m:11026.

    Google Scholar 

  • Neil J. Calkin, Sum-free sets and measure spaces, PhD thesis, Univ. of Waterloo, 1988.

    Google Scholar 

  • Neil J. Calkin, On the number of sum-free sets, Bull. London Math. Soc., 22(1990) 141–144; MR 91b:11015.

    Google Scholar 

  • Neil J. Calkin, On the structure of a random sum-free set of positive integers, Discrete Math., 190(1998) 247–257. 99f:11015

    Google Scholar 

  • Neil J. Calkin & Peter J. Cameron, Almost odd random sum-free sets, Corn-bin. Probab. Comput., 7(1998) 27–32; MR 99c:11012.

    Google Scholar 

  • Neil J. Calkin & Paul Erdös, On a class of aperiodic sum-free sets, Math. Proc. Cambridge Philos. Soc., 120(1996) 1–5; MR 97b:11030.

    Google Scholar 

  • Neil J. Calkin & Steven R. Finch, Conditions on periodicity for sum-free sets, Experiment. Math., 5(1996) 131–137; MR 98c:11010.

    Google Scholar 

  • Neil J Calkin & Jan MacDonald Thomson, Counting generalized sum-free sets, J. Number Theory, 68(1998) 151–159; MR 98m:11010.

    Google Scholar 

  • Peter J. Cameron, Portrait of a typical sum-free set, Surveys in Combinatorics 1987, London Math. Soc. Lecture Notes, 123(1987) Cambridge Univ. Press, 13–42.

    Google Scholar 

  • S. L. G. Choi, On sequences not containing a large sum-free subsequence, Proc. Amer. Math. Soc., 41(1973) 415–418; MR 48 #3910.

    Google Scholar 

  • S. L. G. Choi, On a combinatorial problem in number theory, Proc. London Math. Soc.,(3) 23(1971) 629–642; MR 45 #1867.

    Google Scholar 

  • Louis Funar, Generalized sum-free sets of integers, Nieuw Arch. Wisk. (4) 8(1990) 49–54; MR 91e:11012.

    Google Scholar 

  • Vsevolod Lev, Sharp estimates for the number of sum-free sets, J. reine angew. Math., 555(2003) 1–25.

    MathSciNet  MATH  Google Scholar 

  • Vsevolod Lev, Tomasz Luczak & Tomasz Schoen, Sum-free sets in abelian groups, Israel J. Math., 125(2001) 347–367; MR 2002i:11023.

    Google Scholar 

  • Tomasz Luczak & Tomasz Schoen, On infinite sum-free sets of integers, J. Number Theory, 66(1997) 211–224.

    MathSciNet  MATH  Google Scholar 

  • Tomasz Luczak & Tomasz Schoen, On strongly sum-free subsets of abelian groups, Colloq. Math., 71(1996) 149–151.

    MathSciNet  MATH  Google Scholar 

  • Pieter Moree, On a conjecture of Funar, Nieuw Arch. Wisk.(4) 8(1990) 55–60; MR 91e:11013.

    Google Scholar 

  • Tomasz Schoen, The number of (2,3)-sum-free subsets of {1,..., n}, Acta Arith., 98(2001) 155–163; MR 2002m:1014.

    Google Scholar 

  • Anne Penfold Street, A maximal sum-free set in A5, Utilitas Math., 5(1974) 85–91; MR 49 #7156.

    Google Scholar 

  • Anne Penfold Street, Maximal sum-free sets in abelian groups of order divisible by three, Bull. Austral. Math. Soc., 6(1972) 317–318; MR 47 #5147.

    Google Scholar 

  • P. Varnavides, On certain sets of positive density, J. London Math. Soc., 34(1959) 358–360; MR 21 #5595.

    Google Scholar 

  • Edward T. H. Wang, On double-free set of integers, Ars Combin., 28(1989) 97–100; MR 90d:11011.

    Google Scholar 

  • Yap Hian-Poh, Maximal sum-free sets in finite abelian groups, Bull. Austral. Math. Soc., 4(1971) 217–223; MR 43 #2081 [and see ibid, 5(1971) 43–54; MR 45 #3574; Nanta Math., 2(1968) 68–71; MR 38 #3345; Canad. J. Math., 22(1970) 1185–1195; MR 42 #1897; J. Number Theory, 5(1973) 293–300; MR 48 #11356].

    Google Scholar 

  • N. Alon, & M. Dubiner, Zero-sum sets of prescribed size, Combinatorics, Paul Erdös is eighty, Vol. 1, 33–50, Bolyai Soc. Math. Stud., Janos Bolyai Math. Soc., Budapest, 1993; MR 94j:11016.

    Google Scholar 

  • Noga Alon, Melvyn B. Nathanson & Imre Ruzsa, Adding distinct congruence classes modulo a prime, Amer. Math. Monthly, 102(1995) 250–255; MR 95k:11009.

    Google Scholar 

  • Noga Alon, Melvyn B. Nathanson & Imre Ruzsa, The polynomial method and restricted sums of congruence classes, J. Number Theory, 56(1996) 404–417; MR 96k:11011.

    Google Scholar 

  • A. Bialostocki & P. Dierker, On the Erdös-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings, Discrete Math., 110(1992) 1–8; MR 94a:5147.

    Google Scholar 

  • Arie Bialostocki, Paul Dierker & David Grynkiewicz, On some developments of the Erdös-Ginzburg-Ziv theorem, II, Acta Arith., 110(2003) 173–184.

    MathSciNet  MATH  Google Scholar 

  • A. Bialostocki & M. Lotspeich, Some developments of the Erdös-GinzburgZiv theorem, I, Sets, Graphs é4 Numbers (Budapest, 1991), Colloq. Math. Soc. Janos Bolyai, 60(1992), North-Holland, Amsterdam, 97–117; MR 94k:11021.

    Google Scholar 

  • W. Brakemeier, Ein Beitrag zur additiven Zahlentheorie, Dissertation, Tech. Univ. Braunschweig 1973.

    Google Scholar 

  • W. Brakemeier, Eine Anzahlformel von Zahlen modulon, Monatsh. Math., 85(1978) 277–282.

    MathSciNet  MATH  Google Scholar 

  • Cao Hui-Qin & Sun Zhi-Wei, On sums of distinct representatives, Acta Arith., 87(1998) 159–169; MR 99k:11021.

    Google Scholar 

  • A. L. Cauchy, Recherches sur les nombres, J. École Polytech., 9(1813) 99–116.

    Google Scholar 

  • H. Davenport, On the addition of residue classes, J. London Math. Soc., 10(1935) 30–32.

    Google Scholar 

  • H. Davenport, A historical note, J. London Math. Soc., 22(1947) 100–101.

    MathSciNet  MATH  Google Scholar 

  • J. M. Deshouillers, P. Erdös, S. A. Sârközi, On additive bases, Acta Arith., 30(1976) 121–132; MR 54 #5165.

    Google Scholar 

  • Jean-Marc Deshouillers & Etienne Fouvry, On additive bases, II, J. London Math. Soc.(2), 14(1976) 413–422; MR 56 #2951.

    Google Scholar 

  • J. A. Dias da Silva & Y. O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc., 26(1994) 140–146; see also Linear Algebra Appl., 141(1990) 283–287; MR 91i:15004.

    Google Scholar 

  • Shalom Eliahou & Michel Kervaire, Restricted sums of sets of cardinality 1+p in a vector space over Fp, Discrete Math., 235(2001) 199–213; MR 2002e:11023.

    Google Scholar 

  • R. B. Eggleton & P. Erdös, Two combinatorial problems in group theory, Acta Arith., 21(1972) 111–116; MR 46 #3643.

    Google Scholar 

  • P. Erdös, Some problems in number theory, in Computers in Number Theory, Academic Press, London & New York, 1971, 405–413.

    Google Scholar 

  • P. Erdös, A. Ginzburg & A. Ziv, A theorem in additive number theory, Bull. Res. Council Israel Sect. F Math. Phys., 10F(1961/62) 41–43.

    Google Scholar 

  • P. Erdös & H. Heilbronn, On the addition of residue classes mod p, Acta Arith., 9(1964) 149–159; MR 29 #3463.

    Google Scholar 

  • Carlos Flores & Oscar Ordaz, On the Erdös-Ginzburg-Ziv theorem, Discrete Math., 152(1996) 321–324; MR 97b:05160.

    Google Scholar 

  • G. A. Freiman, Foundations of a Structural Theory of Set Addition, Transl. Math. Monographs, 37(1973), Amer. Math. Soc., Providence RI.

    Google Scholar 

  • Gregory A. Freiman, Lewis Low & Jane Pitman, Sumsets with distinct summands and the Erdös-Heilbronn conjecture on sums of residues. Structure theory of set addition, Astérisque, 258(1999) 163–172; MR 2000j:11036.

    Google Scholar 

  • Luis Gallardo & Georges Grekos, On Brakemeier’s variant of the ErdösGinzberg-Ziz problem, Tatra Mt. Math. Publ., 20(2000) 91–98; MR 2002e:11014.

    Google Scholar 

  • L. Gallardo, G. Grekos, L. Habsieger, B. Landreau & A. Plagne, Restricted addition in Z/nZ and an application to the Erdös-Ginzburg-Ziv theorem, J. London Math. Soc.(2), 65(2002) 513–523; MR 2003b:11011.

    Google Scholar 

  • L. Gallardo, G. Grekos & J. Pihko, On a variant of the Erdös-Ginzburg-Ziv problem, Acta Arith., 89(1999) 331–336; MR 2000k:11026.

    Google Scholar 

  • Gao Wei-Dong, An addition theorem for finite cyclic groups, Discrete Math., 163(1997) 257–265; MR 97k:20039.

    Google Scholar 

  • Gao Wei-Dong, Addition theorems for finite abelian groups, J. Number Theory, 53(1995) 241–246; MR 96i:20070.

    Google Scholar 

  • Gao Wei-Dong, On zero-sum subsequences of restricted size, J. Number Theory, 61(1996) 97–102; II. Discrete Math. 271(2003) 51–59; III. Ars Combin., 61(2001) 65–72; MR 97m:11027; 2003e:11019.

    Google Scholar 

  • Gao Wei-Dong, On the Erdös-Ginsburg-Ziv theorem a survey, Paul Erdös and his mathematics (Budapest, 1999) 76–78, Janos Bolyai Math. Soc., Budapest 1999.

    Google Scholar 

  • Gao Wei-Dong, Note on a zero-sum problem, J. Combin. Theory Ser. A, 95(2001) 387–389; MR 2002i:11016.

    Google Scholar 

  • Gao Wei-Dong & Alfred Geroldinger, On zero-sum sequences in Z/nZ Z/nZ, Electr. J. Combin. Number Theory, 3(2003) A8.

    Google Scholar 

  • Gao Wei-Dong Si Y. O. Hamidoune, Zero sums in abelian groups, Combin. Probab. Comput., 7(1998) 261–263; MR 99h:5118.

    Google Scholar 

  • David J. Grynkiewicz, On four coloured sets with nondecreasing diameter and the Erdös-Ginzburg-Ziv theorem, J. Combin. Theory Ser. A, 100(2002) 44–60; MR 2003i:11032..

    Google Scholar 

  • Y. O. Hamidoune, A. S. Lladô & O. Serra, On restricted sums, Combin. Probab. Comput., 9(2000) 513–518; MR 2002a:20059.

    Google Scholar 

  • Yahya Ould Hamidoune, Oscar Ordaz & Asdrubal Ortuno, On a combinatorial theorem of Erdös, Ginzburg and Ziv, Combin. Probab. Comput., 7(1998) 403–412; MR 2000c:05146.

    Google Scholar 

  • Yahya Ould Hamidoune & Gilles Zémor, On zero-free subset sums, Acta Arith., 78(1996) 143–152; MR 97k:11023.

    Google Scholar 

  • H. Harborth, Ein Extremalproblem für Gitterpunkte, J. reine angew. Math., 262/263(1973) 356–360; MR 48 #6008.

    Google Scholar 

  • N. Hegyvâri, F. Hennecart & A. Plagne, A proof of two Erdös’ conjectures on restricted addition and further results, J. reine angew. Math., 560(2003) 199–220.

    MathSciNet  MATH  Google Scholar 

  • F. Hennecart, Additive bases and subadditive functions, Acta Math. Hungar., 77(1997) 29–40; MR 99a:11013.

    Google Scholar 

  • François Hennecart, Une propriété arithmétique des bases additives. Une critère de non-base, Acta Arith., 108(2003) 153–166; MR 2004c:11012.

    Google Scholar 

  • Hou Qing-Hu & Sun Zhi-Wei, Restricted sums in a field, Acta Arith., 102(2002) 239–249; MR 2003e:11025.

    Google Scholar 

  • A. Kemnitz, On a lattice point problem, Ars Combin., 16(1983) 151–160; MR 85c:52022.

    Google Scholar 

  • J. H. B. Kemperman, On small sumsets in an abelian group, Acta Math., 103(1960) 63–88; MR 22 #1615.

    Google Scholar 

  • Vsevolod F. Lev, Restricted set addition in groups: I. The classical setting, J. London Math. Soc.(2), 62(2000) 27–40; MR 2001g:11024.

    Google Scholar 

  • Li Zheng-Xue, Liu Hui-Qin & Gao Wei-Dong, A conjecture concerning the zero-sum problem, J. Math. Res. Exposition, 21(2001) 619–622; MR 2002i:11018.

    Google Scholar 

  • Liu Jian-Xin & Sun Zhi-Wei, A note on the Erdös-Ginsburg-Ziv theorem, Nanjing Daxue Xuebao Kiran Kexue Ban, 37(2001) 473–476; MR 2002j:11014.

    Google Scholar 

  • Henry B. Mann & John E. Olsen, Sums of sets in the elementary abelian group of type (p, p), J. Combin. Theory, 2(1967) 275–284.

    MATH  Google Scholar 

  • R. Mansfield, How many slopes in a polygon? Israel J. Math., 39(1981) 265–272; MR 84j:3074.

    Google Scholar 

  • Melvyn B. Nathanson, Ballot numbers, alternating products, and the ErdösHeilbronn conjecture, The Mathematics of Paul Erdös, 1199–217, Algorithms Combin., 13 Springer Berlin 1997; MR 971:1008.

    Google Scholar 

  • Melvyn B. Nathanson, Additive number theory. Inverse problems and the geometry of sumsets, Graduate Texts in Mathematics, 165, Springer-Verlag, New York, 1996; MR 98f:11011.

    Google Scholar 

  • M. B. Nathanson & I. Z. Ruzsa, Sums of different residues modulo a prime, 1993 preprint.

    Google Scholar 

  • John E. Olsen, An addition theorem modulo p, J. Combin. Theory, 5(1968) 45–52.

    MathSciNet  Google Scholar 

  • John E. Olsen, An addition theorem for the elementary abelian group, J. Combin. Theory, 5(1968) 53–58.

    Google Scholar 

  • Pan Hao & Sun Zhi-Wei, A lower bound for \(\left| {\left\{ {a + b:a \in A,b \in B,P\left( {a,b} \right)} \right. \ne \left. 0 \right\}} \right|,\) J. Combin. Theory Ser. A 100(2002) 387–393; MR 2003k:11016.

    Google Scholar 

  • J. M. Pollard, A generalisation of the theorem of Cauchy and Davenport, J. London Math. Soc., 8(1974) 460–462.

    MathSciNet  MATH  Google Scholar 

  • J. M. Pollard, Addition properties of residue classes, J. London Math. Soc., 11(1975) 147–152.

    MathSciNet  MATH  Google Scholar 

  • U.-W. Rickert, Ober eine Vermutung in der additiven Zahlentheorie, Dissertation, Tech. Univ. Braunschweig 1976.

    Google Scholar 

  • Oystein J. Rodseth, Sums of distinct residues mod p, Acta Arith., 65(1993) 181–184; MR 94j:11004.

    Google Scholar 

  • Oystein J. Rpdseth, On the addition of residue classes mod p, Monatsh. Math., 121(1996) 139–143; MR 97a:11009.

    Google Scholar 

  • Lajos Rdnyai, On a conjecture of Kemnitz, Combinatorica, 20(2000) 569–573; MR 2001k:11025.

    Google Scholar 

  • C. Ryavec, The addition of residue classes modulon, Pacific J. Math., 26 (1968) 367–373.

    MathSciNet  MATH  Google Scholar 

  • Sun Zhi-Wei, Restricted sums of subsets of Z, Acta Arith., 99(2001) 41–60; MR 2002j:11016.

    Google Scholar 

  • Sun Zhi-Wei, Unification of zero-sum problems, subset sums and covers of Z, Electr. Res. Announcements, Amer. Math. Soc., 9(2003) 51–60.

    Google Scholar 

  • B. Sury & R. Thangadurai, Gao’s conjecture on zero-sum sequences, Proc. Indian Acad. Sci. Math. Sci., 112(2002) 399–414; MR 2003k:11029.

    Google Scholar 

  • E. Szemerédi, On a conjecture of Erdös and Heilbronn, Acta Arith., 17(1970–71) 227–229.

    MathSciNet  MATH  Google Scholar 

  • R. Thangadurai, On a conjecture of Kemnitz, C.R. Math. Acad. Sci. Soc. R. Can., 23(2001) 39–45; MR 2002i:11017.

    Google Scholar 

  • R. Thangadurai, Non-canonical extensions of Erdös-Ginzburg-Ziv theorem, Integers, 2(2002) Paper A7, 14pp.; MR 2003d:11025.

    Google Scholar 

  • R. Thangadurai, Interplay between four conjectures on certain zero-sum problems, Expo. Math., 20(2002) 215–228; MR 2003f:11023.

    Google Scholar 

  • R. Thangadurai, On certain zero-sum problems in finite abelian groups, Current trends in number theory (Allahabad, 2000) 239–254, Hindustan Book Agency, 2002; MR 2003f:11031.

    Google Scholar 

  • Wang Chaot, Note on a variant of the Erdös-Ginzburg-Ziv problem, Acta Arith., 108(2003) 53–59; MR 2004c:11020.

    Google Scholar 

  • H. L. Abbott, On a conjecture of Erdös and Straus on non-averaging sets of integers, Congr. Numer. XV, Proc. 5th Brit. Combin. Conf., Aberdeen, 1975, 1–4; MR 53 #10752.

    Google Scholar 

  • H. L. Abbott, Extremal problems on non-averaging and non-dividing sets, Pacific J. Math., 91(1980) 1–12; MR 82j:10005.

    Google Scholar 

  • H. L. Abbott, On nonaveraging sets of integers, Acta Math. Acad. Sci. Hun-gar., 40(1982) 197–200; MR 84g:10090.

    Google Scholar 

  • H. L. Abbott, On the Erdös—Straus non-averaging set problem, Acta Math. Hungar. 47(1986) 117–119; MR 88e:11010.

    Google Scholar 

  • A. P. Bosznay, On the lower estimation of non-averaging sets, Acta Math. Hungar., 53(1989) 155–157; MR 90d:11016.

    Google Scholar 

  • Fan R. K. Chung & John L. Goldwasser, Integer sets containing no solution to x + y = 3z, The Mathematics of Paul Erd6s, 1218–227, Algorithms Combin., 13 Springer Berlin 1997.

    Google Scholar 

  • P. Erdös & A. Sârközy, On a problem of Straus, Disorder in physical systems, Oxford Univ. Press, New York, 1990, pp. 55–66; MR 911:11012.

    Google Scholar 

  • P. Erdfis & E. G. Straus, Non-averaging sets II, in Combinatorial Theory and its Applications II, Colloq. Math. Soc. Jcinos Bolyai 4, North-Holland, 1970, 405–411; MR 47 #4804.

    Google Scholar 

  • E. G. Straus, Non-averaging sets, Proc. Symp. Pure Math., 19, Amer. Math. Soc., Providence 1971, 215–222.

    Google Scholar 

  • P. Erdös, Some remarks on number theory (Hebrew, English summary), Riveon Lematematika, 9(1955) 45–48; MR 17, 460.

    Google Scholar 

  • Jan Kristian Haugland, Advances in the minimum overlap problem, J. Number Theory, 58(1996) 71–78; MR 99c:11031.

    Google Scholar 

  • M. Katz & F. Schnitzer, On a problem of J. Czipszer, Rend. Sem. Mat. Univ. Padova 44(1970) 85–90; MR 45 #8540.

    Google Scholar 

  • L. Moser, On the minimum overlap problem of Erdös, Acta Arith., 5(1959) 117–119; MR 21 #5594.

    Google Scholar 

  • L. Moser & M. G. Murdeshwar, On the overlap of a function with its translates, Nieuw Arch. Wisk.(3), 14(1966) 15–18; MR 33 #4218.

    Google Scholar 

  • T. S. Motzkin, K. E. Ralston & J. L. Selfridge, Minimum overlappings under translation, Bull. Amer. Math. Soc., 62(1956) 558.

    Google Scholar 

  • S. Swierczkowski, On the intersection of a linear set with the translation of its complement, Colloq. Math., 5(1958) 185–197; MR 21 #1955.

    Google Scholar 

  • W. Ahrens, Mathematische Unterhaltungen and Spiele, Vol. 1, B. G. Teubner, Leipzig 1921.

    Google Scholar 

  • Claude Berge, Theory of Graphs, Methuen, 1962, p. 41.

    Google Scholar 

  • Paul Berman, Problem 122, Pi Mu Epsilon J., 3(1959–64) 118, 412.

    Google Scholar 

  • A. Bruen & R. Dixon, The n-queens problem, Discrete Math., 12(1975) 393–395; MR 51 #12555.

    Google Scholar 

  • E. J. Cockayne & S. T. Hedetniemi, A note on the diagonal queens domination problem, DM-301-IR, Univ. of Victoria, BC, March 1984.

    Google Scholar 

  • B. Gamble, B. Shepherd & E. T. Cockayne, Domination of chessboards by queens on a column, DM-318-IR, Univ. of Victoria, BC, July 1984.

    Google Scholar 

  • Solomon W. Golomb & Herbert Taylor, Constructions and properties of Costas arrays, Dept. Elec. Eng., Univ. S. California, July 1981-Oct. 1983.

    Google Scholar 

  • B. Hansche & W. Vucenic, On the n-queens problem, Notices Amer. Math. Soc., 20(1973) A-568.

    Google Scholar 

  • Olof Heden, On the modular n-queens problem, Discrete Math., 102(1992) 155–161; MR 93d:05041; II, 142(1995) 97–106; MR 96e:51007; III, 243(2002) 135–150; MR 2002m:51003.

    Google Scholar 

  • G. B. Huff, On pairings of the first 2n natural numbers, Acta. Arith., 23(1973) 117–126.

    MathSciNet  MATH  Google Scholar 

  • D. A. Klarner, The problem of the reflecting queens, Amer. Math. Monthly 74(1967) 953–955; MR 40 #7123.

    Google Scholar 

  • Torleiv K10ve, The modular n-queen problem, Discrete Math., 19(1977) 289–291; MR 57 #2952; II, 36(1981) 33–48; MR 82k:5034.

    Google Scholar 

  • Maurice Kraitchik, Mathematical Recreations, Norton, New York, 1942, 247–256.

    Google Scholar 

  • Scott P. Nudelman, The modular n-queens problem in higher dimensions, Discrete Math., 146(1995) 159–167; MR 97b:11028.

    Google Scholar 

  • Patric R. J. Ostergârd & William D. Weakley, Values of domination numbers of the queen’s graph, Electron. J. Combin., 8(2001) R29, 19pp.; MR 2002f:05122.

    Google Scholar 

  • Igor Rivin, Ilan Vardi & Paul Zimmermann, The n-queens problem, Amer. Math. Monthly, 101(1994) 629–639; MR 95d:05009.

    Google Scholar 

  • J. D. Sebastian, Some computer solutions to the reflecting queens problem, Amer. Math. Monthly, 76(1969) 399–400; MR 39 #4018.

    Google Scholar 

  • J. L. Selfridge, Pairings of the first 2n integers so that sums and differences are all distinct, Notices Amer. Math. Soc., 10(1963) 195.

    Google Scholar 

  • Shen Mok-Kong & Shen Tsen-Pao, Research Problem 39, Bull. Amer. Math. Soc., 68 (1962) 557.

    MathSciNet  Google Scholar 

  • B. Shepherd, B. Gamble & E. T. Cockayne, Domination parameters for the bishops graph, DM-327-IR, Univ. of Victoria, BC, Oct. 1884.

    Google Scholar 

  • M. Slater, Problem 1, Bull. Amer. Math. Coc., 69(1963) 333.

    Google Scholar 

  • P. H. Spencer & E. J. Cockayne, An upper bound for the domination number of the queens graph, DM-376-IR, Univ. of Victoria, BC, June 1985.

    Google Scholar 

  • Ilan Vardi, Computational Recreations in Mathematica©, Addison-Wesley, Redwood City CA, 1991, Chap. 6.

    Google Scholar 

  • G. E. Andrews, ETPHKA! num = A + A + A, J. Number Theory, 23(1986) 285–293.

    MathSciNet  MATH  Google Scholar 

  • Paul T. Bateman, Adolf Hildebrand & George B. Purdy, Expressing a positive integer as a sum of a given number of distinct squares of positive integers, Acta Arith., 67(1994)349–380; MR 95j:11092.

    Google Scholar 

  • Claus Bauer, A note on sums of five almost equal prime squares, Arch. Math. (Basel) 69(1997) 20–30; MR 98c:11110.

    Google Scholar 

  • Claus Bauer, Liu Ming-Chit & Zhan Tao, On a sum of three prime squares, J. Number Theory), 85(2000) 336–359; MR 2002e:11135.

    Google Scholar 

  • Jan Bohman, Carl-Erik Fröberg & Hans Riesel, Partitions in squares, BIT, 19(1979) 297–301; see Gupta’s MR 80k:10043.

    Google Scholar 

  • Jörg Brüdern & Etienne Fouvry, Lagrange’s four squares theorem with almost prime variables, J. reine angew. Math., 454(1994) 59–96; MR 96e:11125.

    Google Scholar 

  • J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set, Acta Sci. Math. Szeged, 21(1960) 111–124; MR 24 #A103.

    Google Scholar 

  • Chan Heng-Huat & Chua Kok-Seng, Representations of integers as sums of 32 squares, Ramanujan J., (to appear)

    Google Scholar 

  • Noam Elkies & Irving Kaplansky, Problem 10426, Amer. Math. Monthly, 102(1995) 70; solution, Andrew Adler, 104(1997) 574.

    Google Scholar 

  • John A. Ewell, On sums of triangular numbers and sums of squares, Amer. Math. Monthly, 99(1992) 752–757; MR 93j:11021.

    Google Scholar 

  • Andrew Granville & Zhu Yi-Liang, Representing binomial coefficients as sums of squares, Amer. Math. Monthly 97(1990) 486–493; MR 92b:11009.

    Google Scholar 

  • Emil Grosswald & L. E. Mattics, Solutions to problem E 3262, Amer. Math. Monthly, 97(1990) 240–242.

    MathSciNet  Google Scholar 

  • F. Halter-Koch, Darstellung natürliche Zahlen als Summe von Quadraten, Acta Arith., 42(1982) 11–20; MR 84b:10025.

    Google Scholar 

  • D. R. Heath-Brown & D. I. Tolev, Lagrange’s four squares theorem with one prime and three almost-prime variables, J. reine angew. Math., 558(2003) 159–224; MR 2004c:11180.

    Google Scholar 

  • Michael D. Hirschhorn, On partitions into four distinct squares of equal parity, Australas. J. Combin., 24(2001) 285–291; MR 2002f:05019.

    Google Scholar 

  • Michael D. Hirschhorn, On sums of squares, Austral. Math. Soc. Gaz., 29(2002) 86–87.

    MathSciNet  Google Scholar 

  • Michael D. Hirschhorn, Comment on `Dissecting squares’, Note 87.33 Math. Gaz., 87(Jul 2003) 286–287.

    Google Scholar 

  • Michael D. Hirschhorn & James A. Sellers, On representations of a number as a sum of three triangles, Acta Arith., 77(1996) 289–301; MR 97e:11119.

    Google Scholar 

  • Michael D. Hirschhorn & James A. Sellers, Some relations for partitions into four squares, Special functions (Hong Kong, 1999) 118–124, World Sci. Publishing, River Edge NJ, 2000; MR 2001k:11201.

    Google Scholar 

  • James G. Huard & Kenneth S. Williams, Sums of sixteen squares, Far East J. Mat. Sci., 7(2002) 147–164; MR 2003m:11059.

    Google Scholar 

  • James G. Huard & Kenneth S. Williams, Sums of twelve squares, Acta Arith., 109(2003) 195–204; MR 2004a:11029.

    Google Scholar 

  • Lin Jia-Fu, The number of representations of an integer as a sum of eight triangular numbers, Chinese Quart. J. Math., 15(2000) 66–68; MR 2002e:11049.

    Google Scholar 

  • Liu Jian-Ya & Zhan Tao, On sums of five almost equal prime squares, Acta Arith., 77(1996) 369–383; MR 97g:11113; II, Sci. China Ser., 41(1998) 710–722; MR 99h:11115.

    Google Scholar 

  • Stephen C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions. Reprinted from Ramanujan J., 6(2002) 7–149; MR 2003m:11060. With a preface by George E. Andrews. Developments in Mathematics, 5, Kluwer Academic Publishers, Boston MA, 2002; see also Proc. Nat. Acad. Sci. U.S.A., 93(1996) 15004–15008; MR 99k:11162.

    Google Scholar 

  • Hugh L. Montgomery & Ulrike M. A. Vorhauer, Greedy sums of distinct squares, Math. Comput., 73(2004) 493–513.

    MathSciNet  MATH  Google Scholar 

  • G. Pall, On sums of squares, Amer. Math. Monthly, 40(1933) 10–18.

    MathSciNet  MATH  Google Scholar 

  • V. A. Plaksin, Representation of numbers as a sum of four squares of integers, two of which are prime, Soviet Math. Dokl., 23(1981) 421–424; transl. of Dokl. Akad. Nauk SSSR, 257(1981) 1064–1066; MR 82h:10027.

    Google Scholar 

  • Stefan Porubskÿ, Sums of prime powers, Monatsh. Math., 86(1979) 301–303.

    MATH  Google Scholar 

  • A. Schinzel & W. Sierpinski, Acta Arith., 4(1958) 185–208; erratum, 5(1959) 259; MR 21 #4936.

    Google Scholar 

  • W. Sierpinski, Sur les nombres triangulaires qui sont sommes de deux nombres triangulaires, Elem. Math., 17(1962) 63–65.

    Google Scholar 

  • Goro Shimura, The representation of integers as sums of squares, Amer. J. Math., 124(2002) 1059–1081; MR 20031:11049.

    Google Scholar 

  • R. P. Sprague, Über zerlegungen in ungleiche Quadratzahlen, Math. Z., 51 (1949) 289–290; MR 10 283d.

    Google Scholar 

  • E. M. Wright, The representation of a number as a sum of five or more squares, Quart. J. Math. Oxford, 4(1933) 37–51, 228–232.

    MATH  Google Scholar 

  • E. M. Wright, The representation of a number as a sum of four `almost proportional’ squares, Quart. J. Math. Oxford, 7(1936) 230–240.

    Google Scholar 

  • E. M. Wright, Representation of a number as a sum of three or four squares, Proc. London Math. Soc.(2), 42(1937) 481–500.

    Google Scholar 

  • Jörg Brüdern, Sums of squares and higher powers, I. II. J. London Math. Soc. (2), 35(1987) 233–243, 244–250; MR 88f:11096.

    Google Scholar 

  • Jörg Brüdern, A problem in additive number theory, Math. Proc. Cambridge Philos. Soc., 103(1988) 27–33; MR 89c:11150.

    Google Scholar 

  • Jörg Brüdern, Ann. Sci. École Norm. Sup., 28(1995) 461–476; MR 96e:11126.

    Google Scholar 

  • Stephan Daniel, On gaps between numbers that are sums of three cubes, Mathematika, 44(1997) 1–13; MR 98c:11105.

    Google Scholar 

  • Kevin B. Ford, The representation of numbers as sums of unlike powers, J. London Math. Soc. (2), 51(1995) 14–26; MR 96c:11115.

    Google Scholar 

  • Kevin B. Ford, The representation of numbers as sums of unlike powers, II, J. Amer. Math. Soc., 9(1995) 919–940; MR 97g:11111.

    Google Scholar 

  • R. L. Graham, Complete sequences of polynomial values, Duke Math. J., 31(1964) 275–285; MR 29 #63.

    Google Scholar 

  • Koichi Kawada, Note on the sums of cubes of primes and an almost prime, Arch. Math. (Basel) 69(1997) 13–19; MR 98f:11105.

    Google Scholar 

  • Lin Shen, Computer experiments on sequences which form integral bases, in J. Leech (editor) Computational Problems in Abstract Algebra, 365–370, Pergamon, 1970.

    Google Scholar 

  • Cameron Douglas Patterson, The Derivation of a High Speed Sieve Device, PhD thesis, The Univ. of Calgary, March 1992.

    Google Scholar 

  • H. E. Richert, Uber zerlegungen in paarweise verschiedene zahlen, Norsk Mat. Tidskrift, 31(1949) 120–122.

    MathSciNet  MATH  Google Scholar 

  • Fernando Rodriguez Villegas & Don Zagier, Which primes are sums of two cubes? Number Theory (Halifax NS, 1994) 295–306, CMS Conf. Proc., 15, Amer. Math. Soc., 1995.

    Google Scholar 

  • R. P. Sprague, Uber zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z., 51(1948) 466–468; MR 10 514h.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, T2N 1N4, Calgary, Alberta, Canada

    Richard K. Guy

Authors
  1. Richard K. Guy
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Guy, R.K. (2004). Additive Number Theory. In: Unsolved Problems in Number Theory. Problem Books in Mathematics, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-0-387-26677-0_4

Download citation

  • DOI: https://doi.org/10.1007/978-0-387-26677-0_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-1928-1

  • Online ISBN: 978-0-387-26677-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation