Abstract
We can partition the positive integers into three classes:
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the unit 1
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the primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...
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the composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16, ...
A number greater than 1 is prime if its only positive divisors are 1 and itself; otherwise it’s composite. Primes have interested mathematicians at least since Euclid, who showed that there are infinitely many. The largest prime in the Bible is 22273 at Numbers, 3 xliii.
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References
William Adams and Daniel Shanks, Strong primality tests that are not sufficient, Math. Comput.,39(1982) 255–300.
Manindra Agrawal, Neeraj Kayal and Nitin Saxena, Primes is in P, Annals of Math.,(to appear)
F. Arnault, Rabin-Miller primality test: composite numbers which pass it, Math. Comput., 64 (1995) 355–361.
Stephan Baier, On the Bateman-Horn conjecture, J. Number Theory, 96 (2002) 432–448.
Friedrich L. Bauer, Why Legendre made a wrong guess about ir(x), and how Laguerre’s continued fraction for the logarithmic integral improved it, Math. Intelligencer, 25 (2003) 7–11.
Folkmar Bornemann, PRIMES is in P: a breakthrough for “Everyman”, Notices Amer. Math. Soc., 50(2003) 545–552; MR 2004c: 11237.
Henri Cohen, Computational aspects of number theory, Mathematics unlimited — 2001 and beyond, 301–330, Springer, Berlin, 2001; MR 2002i: 11126.
Matthew M. Conroy, A sequence related to a conjecture of Schinzel, J. Integer Seq., 4(2001) no.1 Article 01.1.7; MR 2002j: 11017.
Richard E. Crandall and Carl Pomerance, Prime Numbers: a computational perspective, Springer, New York, 2001; MR 2002a: 11007.
Marc Deléglise and Joel Rivat, Computing 7r(x): the Meissel, Lehmer, Lagarias, Miller, Odlyzko method, Math. Comput.,65(1996) 235–245; MR 96d:11139.
L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers, Messenger of Mathematics, 33 (1904) 155–161.
Pierre Dusart, The kth prime is greater than k(ln k + ln ln k — 1) for k 2, Math. Comput., 68(1999) 411–415; MR 99d: 1 1133.
P. D. T. A. Elliott, On primes and powers of a fixed integer, J. London Math. Soc. (2), 67(2003) 365–379; MR 2003m: 11151.
Andrew Granville, Unexpected irregularities in the distribution of prime numbers, Proc. Internat. Congr. Math., Vol. 1, 2 (Zurich, 1994 ), 388–399.
Andrew Granville, Prime possibilities and quantum chaos, Emissary,Spring 2002, ppl, 12–18.
Wilfrid Keller, Woher kommen die größten derzeit bekannten Primzahlen? Mitt. Math. Ges. Hamburg, 12(1991) 211–229; MR 92j: 1 1006.
J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing 7r(x): the MeisselLehmer method, Math. Comput., 44(1985) 537–560; MR 86h: 1 1111.
J. C. Lagarias and A. M.Odlyzko, Computing 7r(x): an analytic method, J. Algorithms, 8(1987) 173–191; MR 88k: 1 1095.
Arjen K. Lenstra and Mark S. Manasse, Factoring by electronic mail, in Advances in Cryptology EUROCRYPT’89, Springer Lect. Notes in Comput. Sci., 434(1990) 355–371; MR 91i: 1 1182.
Hendrik W. Lenstra, Factoring integers with elliptic curves, Ann. of Math. (2), 126(1987) 649–673; MR 89g: 1 1125.
Hendrik W. Lenstra and Carl Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc., 5(1992) 483–916; MR 92m: 1 1145.
Richard F. Lukes, Cameron D. Patterson and Hugh Cowie Williams, Numerical sieving devices: their history and some applications, Nieuw Arch. Wisk.(4), 13(1995) 113–139; MR 99m: 1 1082.
James K. McKee, Speeding Fermat’s factoring method, Math. Comput., 68(1999) 1729–1737; MR 2000b: 11138.
G. L. Miller, Riemann’s hypothesis and tests for primality, J. Comput. System Sci., 13(1976) 300–317; MR 58 #470ab.
Peter Lawrence Montgomery, An FFT extension of the elliptic curve method of factorization, PhD dissertation, UCLA, 1992.
Siguna Müller, A probable prime test with very high confidence for n = 3 mod 4, J. Cryptology, 16 (2003) 117–139.
J. M. Pollard, Theorems on factoring and primality testing, Proc. Cambridge Philos. Soc., 76(1974) 521–528; MR 50 #6992.
J. M. Pollard, A Monte Carlo method for factorization, BIT,15(1975) 331–334; MR 52 #13611.
Carl Pomerance, Recent developments in primality testing, Math. Intelligencer, 3(1980/81) 97–105.
Carl Pomerance, Notes on Primality Testing and Factoring, MAA Notes 4(1984) Math. Assoc. of America, Washington DC.
Carl Pomerance (editor), Cryptology and Computational Number Theory, Proc. Symp. Appl. Math., 42 Amer. Math. Soc., Providence, 1990; MR 91k: 1 1113.
Paulo Ribenboim, The Book of Prime Number Records, Springer-Verlag, New York, 1988.
Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, New York, 1991.
Hans Riesel, Prime Numbers and Computer Methods for Factorization,2nd ed, Birkhäuser, Boston, 1994; MR 95h:11142.
Hans Riesel, Wie schnell kann man Zahlen in Faktoren zerlegen? Mitt. Math. Ges. Hamburg, 12 (1991) 253–260.
R. Rivest, A. Shamir and L. Adleman, A method for obtaining digital signatures and public key cryptosystems, Communications A.C.M., Feb. 1978.
Michael Rubinstein and Peter Sarnak, Chebyshev’s bias, Experimental Math., 3(1994) 173–197; MR 96d: 1 1099.
A. Schinzel and W. Sierpinski, Sur certaines hypothèses concernant les nombres premiers, Acta Arith.,4(1958) 185–208; erratum 5(1959) 259; MR 21 #4936; and see 7(1961) 1–8; MR 24 #A70.
R. Solovay and V. Strassen, A fast Monte-Carlo test for primality, SIAM J. Comput.,6(1977) 84–85; erratum 7(1978) 118; MR 57 #5885.
Michael Somos and Robert Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(2003) 539–540; MR 2004c: 11007.
Jonathan Sorenson, Counting the integers cyclotomic methods can factor, Comput. Sci. Tech. Report, 919, Univ. of Wisconsin, Madison, March 1990.
Hugh Cowie Williams, Édouard Lucas and Primality Testing, Canad. Math. Soc. Monographs 22, Wiley, New York, 1998.
H. C. Williams and J. S. Judd, Some algorithms for prime testing using generalized Lehmer functions, Math. Comput., 30 (1976) 867–886.
Alberto Barajas and Rita Zuazua, A sieve for primes of the form n2+1, preprint, Williams60, Banff, May 2003.
Chen Yong-Gao, Gabor Kun, Gabor Pete, Imre Z. Ruzsa and Adam Timar, Prime values of reducible polynmials II, Acta Arith., 04 (2002) 117–127.
Cécile Dartyge, Le plus grand facteur de n2 + 1, où n est presque premier, Acta Arith., 76 (1996) 199–226.
Cécile Dartyge, Entiers de la forme n2 + 1 sans grand facteur premier, Acta Math. Hungar., 72 (1996) 1–34.
Cécile Dartyge, Greg Martin and Gérald Tenenbaum, Polynomial values free of large prime factors, Period. Math. Hungar., 43(2001) 111–119; MR 2002c: 11117.
Etienne Fouvry and Henryk Iwaniek, Gaussian primes, Acta Arith., 79(1997) 249–287; MR 98a: 1 1120.
John Friedlander and Henryk Iwaniec, The polynomial X2 + Y4 captures its primes. Ann. of Math.(2), 148(1998) 945–1040; MR 2000c: 1 1150a.
Gilbert W. Fung and Hugh Cowie Williams, Quadratic polynomials which have a high density of prime values, Math. Comput., 55(1990) 345–353; MR 90j: 1 1090.
Martin Gardner, The remarkable lore of prime numbers, Scientific Amer.,210 #3 (Mar. 1964) 120–128.
G. H. Hardy and J. E. Littlewood, Some problems of ‘partitio numerorum’ III: on the expression of a number as a sum of primes, Acta Math., 44 (1923) 1–70.
D. R. Heath-Brown, The Pyateckii-Sapiro prime number theorem, J. Number Theory, 16 (1983) 242–266.
D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc.(3) 64(1992) 265338.
D. Roger Heath-Brown, Primes represented by x3+2y3, Acta Math., 186(2001) 1–84; MR 2002b: 11122.
D. Roger Heath-Brown and B. Z. Moroz, Primes represented by binary cubic forms, Proc. London Math. Soc.(3), 84 (2002) 257–288.
Henryk Iwaniec, Almost-primes represented by quadratic polynomials, Invent. Math., 47(1978) 171–188; MR 58 #5553.
M. J. Jacobson, Computational techniques in quadratic fields, M.Sc thesis, Univ. of Manitoba, 1995.
Michael J. Jacobson Jr. and Hugh C. Williams, New quadratic polynomials with high densities of prime values, Math. Comput., 72(2003) 499–519; MR 2003k: 11146.
G. A. Kolesnik, The distribution of primes in sequences of the form [ne], Mat. Zametki(2), 2 (1972) 117–128.
G. A. Kolesnik, Primes of the form [ne], Pacific J. Math. (2), 118 (1985) 437–447.
D. Leitmann, Abschätzung trigonometrischer Summen, J. reine angew. Math., 317 (1980) 209–219.
D. Leitmann, Durchschnitte von Pjateckij-Shapiro-Folgen, Monatsh. Math., 94 (1982) 33–44.
H. Q. Liu and J. Rivat, On the Pyateckii-Sapiro prime number theorem, Bull. London Math. Soc., 24 (1992) 143–147.
Maurice Mignotte, P(x2 + 1) 17 si x 240, C. R. Acad. Sci. Paris Sér. I Math., 301(1985) 661–664; MR 87a: 1 1026.
Carl Pomerance, A note on the least prime in an arithmetic progression, J. Number Theory, 12 (1980) 218–223.
I. I. Pyateckii-Sapiro, On the distribution of primes in sequences of the form [1(n)] (Russian), Mat. Sbornik N.S., 33(1953) 559–566; MR 15, 507.
Joel Rivat and Patrick Sargos, Nombres premiers de la forme [ne], Canad. J. Math., 53(2001) 414–433: MR 2002a: 11107.
J. Rivat and J. Wu, Prime numbers of the form [ne], Glasg. Math. J., 43(2001) 237–254; MR 2002k: 11154.
Daniel Shanks, On the conjecture of Hardy and Littlewood concerning the number of primes of the form n2 + a, Math. Comput., 14 (1960) 321–332.
W. Sierpinski, Les binômes x2 + n et les nombres premiers, Bull. Soc. Roy. Sci. Liège, 33 (1964) 259–260.
E. R. Sirota, Distribution of primes of the form p = [ne] = t d in arithmetic progressions (Russian), Zap. Nauchn. Semin. Leningrad Otdel. Mat. Inst. Steklova, 121(1983) 94–102; Zbl. 524. 10038.
Panayiotis G. Tsangaris, A sieve for all primes of the form x2 + (x +1) 2, Acta Acad. Paedagog. Agriensis Sect. Mat. (N.S.) bf25(1998) 39–53 (1999); MR 2001c: 11044.
Takashi Agoh, Karl Dilcher and Ladislav Skula, Wilson quotients for composite moduli, Math. Comput., 67(1998) 843–861; MR 98h: 1 1003.
I. O. Angell and H. J. Godwin, Some factorizations of 10’ + 1, Math. Comput., 28 (1974) 307–308.
Chris K. Caldwell and Yves Gallot, On the primality of n! f 1 and 2 x 3 x 5 x... x p + 1, Math. Comput.,71(2002) 441–448; MR 2002g:11011.
Alan Borning, Some results for k! + 1 and 2. 3. 5... p f 1, Math. Comput., 26 (1972) 567–570.
J. P. Buhler, R. E. Crandall and M. A. Penk, Primes of the form n! + 1 and 2.3.5... p f 1, Math. Comput.,38(1982) 639–643; corrigendum, Wilfrid Keller, 40(1983) 727; MR 83c:10006, 85b:11119.
Chris K. Caldwell, On the primality of n! + 1 and 2. 3. 5... p f 1, Math. Comput.,64(1995) 889–890; MR 95g:11003.
Chris K. Caldwell and Harvey Dubner, Primorial, factorial and multifactorial primes, Math. Spectrum,26(1993–94) 1–7.
Chris Caldwell and Yves Gallot, On the primality of n! f 1 and 2 x 3 x 5 x . . . x p + 1, Math. Comput.,71(2002) 441–448.
Harvey Dubner, Factorial and primorial primes, J. Recreational Math., 19 (1987) 197–203.
Martin Gardner, Mathematical Games, Sci. Amer., 243#6(Dec. 1980 ) 18–28.
Solomon W. Golomb, The evidence for Fortune’s conjecture, Math. Mag., 54 (1981) 209–210.
G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly, 109 (2002) 554–559.
S. Kravitz and D. E. Penney, An extension of Trigg’s table, Math. Mag., 48 (1975) 92–96.
Le Mao-Hua and Chen Rong-Ji, A conjecture of Erd6s and Stewart concerning factorials, J. Math. (Wuhan) 23 (2003) 341–344.
S. S. Pillai, Question 1490, J. Indian Math. Soc., 18 (1930) 230.
Mark Templer, On the primality of k! + 1 and 2 * 3 * 5 *... * p + 1, Math. Comput.,34(1980) 303–304.
Miodrag Zivkovie, The number of primes Ez 1(-1)’-’i! is finite, Math. Comput., 68(1999) 403–409; MR 99c: 1 1163.
Takashi Agoh, On Fermat and Wilson quotients, Exposition. Math., 14(1996) 145–170; MR 97e: 1 1008.
Takashi Agoh, Karl Dilcher and Ladislav Skula, Fermat quotients for composite moduli, J. Number Theory, 66(1997) 29–50; MR 98h: 1 1002.
R. C. Archibald, Scripta Math., 3 (1935) 117.
A. O. L. Atkin and N. W. Rickert, Some factors of Fermat numbers, Abstracts Amer. Math. Soc., 1 (1980) 211.
P. T. Bateman, J. L. Selfridge and S. S. Wagstaff, The new Mersenne conjecture, Amer. Math. Monthly, 96(1989) 125–128; MR 90c: 1 1009.
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comput., 67(1998) 441–446; MR 98e: 1 1008.
Wieb Bosma, Explicit primality criteria for h.2 k +1, Math. Comput., 61 (1993) 97–109.
R. P. Brent, Factorization of the eleventh Fermat number, Abstracts Amer. Math. Soc.,10(1989) 89T-11–73.
R. P. Brent, R. E. Crandall, K. Dilcher and C. van Halewyn, Three new factors of Fermat numbers, Math. Comput., 69(2000) 1297–1304; MR 2000j: 11194.
R. P. Brent and J. M. Pollard, Factorization of the eighth Fermat number, Math. Comput., 36(1981) 627–630; MR 83h:10014.
John Brillhart and G. D. Johnson, On the factors of certain Mersenne numbers, I, II Math. Comput.,14(1960) 365–369, 18(1964) 87–92; MR 23#A832, 28#2992.
John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman and Samuel S. Wagstaff, Factorizations of b’ + 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers, Contemp. Math.,22. Amer. Math. Soc., Providence RI, 1983, 1988; MR 84k:10005, 90d:11009.
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comput.,50(1988) 251–260 and S1—S15.
John Brillhart, J. Tonascia and P. Weinberger, On the Fermat quotient, in Computers in Number Theory, Academic Press, 1971, 213–222.
Chris K. Caldwell and Yves Gallot, On the primality of n! f 1 and 2 x 3 x 5 x . . . x p f 1, Math. Comput., 71(2002) 441–448 (electronic).
W. N. Colquitt and L. Welsh, A new Mersenne prime, Math. Comput., 56(1991) 867–870; MR 91h: 1 1006.
Curtis Cooper and Manjit Parihar, On primes in the Fibonacci and Lucas sequences, J. Inst. Math. Comput. Sci., 15 (2002) 115–121.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comput., 66(1997) 433–449; MR 97c: 1 1004.
Richard E. Crandall, J. Doenias, C. Norrie and Jeff Young, The twenty-second Fermat Number is composite, Math. Comput., 64(1995) 863–868; MR 95f: 1 1104.
Richard E. Crandall, Ernst W. Mayer and Jason S. Papadopoulos, The twenty-fourth Fermat number is composite, Math. Comput., 72(2003) 1555–1572; MR 2004c: 11236.
Karl Dilcher, Fermat numbers, Wieferich and Wilson primes: computations and generalizations, Public-key cryptography and computational number theory (Warsaw 2000) 29–48, de Gruyter, Berlin, 2001; MR 2002j: 11004.
Harvey Dubner, Generalized Fermat numbers, J. Recreational Math.,18 (1985–86) 279–280.
Harvey Dubner, Generalized repunit primes, Math. Comput., 61 (1993) 927–930.
Harvey Dubner, Repunit R49081 is a probable prime, Math. Comput., 71(2002) 833–835; MR 2002k: 11224.
Harvey Dubner and Yves Gallot, Distribution of generalized Fermat prime numbers, Math. Comput., 71(2002) 825–832; MR 2002j: 11156.
Harvey Dubner and Wilfrid Keller, Factors of generalized Fermat numbers, Math. Comput., 64 (1995) 397–405; MR 95c: 1010.
Harvey Dubner and Wilfrid Keller, New Fibonacci and Lucas primes, Math. Comput., 68(1999) 417–427, S1—S12; MR 99c11008.
John R. Ehrman, The number of prime divisors of certain Mersenne numbers, Math. Comput., 21(1967) 700–704; MR 36#6368.
Peter Fletcher, William Lindgren and Carl Pomerance, Symmetric and asymmetric primes, J. Number Theory, 58(1996) 89–99; MR 97c: 1 1007.
Donald B. Gillies, Three new Mersenne primes and a statistical theory, Math. Comput., 18(1964) 93–97; MR 28#2990.
Gary B. Gostin, New factors of Fermat numbers, Math. Comput., 64(1995) 393–395; MR 95c: 1 1151.
Gary B. Gostin and Philip B. McLaughlin, Six new factors of Fermat numbers, Math. Comput., 38(1982) 645–649; MR 83c:10003.
R. L. Graham, A Fibonacci-like sequence of composite numbers, Math. Mag., 37(1964) 322–324; Zbl 125, 21.
Richard K. Guy, The strong law of small numbers, Amer. Math. Monthly 95(1988) 697–712; MR 90c:11002 (see also Math. Mag., 63(1990) 3–20; MR 91a: 1 1001.
Kazuyuki Hatada, Mod 1 distribution of Fermat and Fibonacci quotients and values of zeta functions at 2 — p, Comment. Math. Univ. St. Paul. 36(1987), no. 1, 41–51; MR 88i: 1 1085.
G. L. Honaker and Chris K. Caldwell, Palindromic prime pyramids, J. Recreational Math., 30(1999–2000) 169–176.
A. Grytczuk, M. Wôjtowicz and Florian Luca, Another note on the greatest prime factors of Fermat numbers, Southeast Asian Bull. Math., 25(2001) 111115; MR 2002g: 11008.
Wilfrid Keller, Factors of Fermat numbers and large primes of the form k.2n + 1, Math. Comput., 41(1983) 661–673; MR 85b: 1 1117.
Wilfrid Keller, New factors of Fermat numbers, Abstracts Amer. Math. Soc., 5 (1984) 391
Wilfrid Keller, The 17th prime of the form 5.2’ + 1, Abstracts Amer. Math. Soc., 6 (1985) 121.
Christoph Kirfel and Oystein J. Rodseth, On the primality of wh 3h + 1, Selected papers in honor of Helge Tverberg, Discrete Math., 241 (2001) 395–406.
Joshua Knauer and Jörg Richstein, The continuing search for Weiferich primes, Math. Comput., (2004)
Donald E. Knuth, A Fibonacci-like sequence of composite numbers, Math. Mag., 63(1990) 21–25; MR 91e: 1 1020.
M. Kraitchik, Sphinx, 1931, 31.
Michal Krfzek and Jan Chleboun, Is any composite Fermat number divisible by the factor 5h2’’1 + 1, Tatra Mt. Math. Publ., 11(1997) 17–21; MR 98j: 1 1003.
Michal Kffzek and Lawrence Somer, A necessary and sufficient condition for the primality of Fermat numbers, Math. Bohem., 126(2001) 541–549; MR 2004a: 11004.
D. H. Lehmer, Sphinx, 1931, 32, 164.
D. H. Lehmer, On Fermat’s quotient, base two, Math. Comput., 36(1981) 289–290; MR 82e:10004.
A. K. Lenstra, H. W. Lenstra, M. S. Manasse and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comput., 61(1993) 319–349; MR 93k: 1 1116.
J. C. P. Miller and D. J. Wheeler, Large prime numbers, Nature, 168 (1951) 838.
Satya Mohit and M. Ram Murty, Wieferich primes and Hall’s conjecture, C.R. Math. Acad. Sci. Soc. Roy. Can., 20(1998) 29–32; MR 98m: 1 1004.
Peter L. Montgomery, New solutions of aP - 1-1 mod p 2, Math. Comput., 61(1993) 361–363; MR 94d: 1 1003.
Pieter Moree and Peter Stevenhagen, Prime divisors of the Lagarias sequence, J. Théor. Nombres Bordeaux, 13(2001) 241–251; MR 2002c: 11016.
Thorkil Naur, New integer factorizations, Math. Comput., 41(1983) 687–695; MR 85c: 1 1123.
Rudolf Ondrejka, Titanic primes with consecutive like digits, J. Recreational Math., 17(1984–85) 268–274.
Pi Xin-Ming, Generalized Fermat primes for b 2000, m 10, J. Math. (Wuhan) 22(2002) 91–93; MR 2003a: 11008.
Paulo Ribenboim, On square factors of terms of binary recurring sequences and the ABC conjecture, Publ. Math. Debrecen, 59(2001) 459–469; MR 2002i: 11033.
Herman te Riele, Walter Lioen and Dik Winter, Factorization beyond the googol with MPQS on a single computer, CWI Quarterly, 4 (1991) 69–72.
Hans Riesel Si Anders Bjorn, Generalized Fermat numbers, in Mathematics of Computation 1943–1993, Proc. Symp. Appl. Math.,48(1994) 583–587; MR 95j:11006.
Raphael M. Robinson, Mersenne and Fermat numbers, Proc. Amer. Math. Soc., 5(1954) 842–846; MR 16, 335.
A. Rotkiewicz, Note on the diophantine equation 1 + x + x2 +… + xn = y nz, Elem. Math., 42 (1987) 76.
Daniel Shanks and Sidney Kravitz, On the distribution of Mersenne divisors, Math. Comput., 21(1967) 97–100; MR 36:#3717.
Hiromi Suyama, Searching for prime factors of Fermat numbers with a microcomputer (Japanese) bit, 13(1981) 240–245; MR 82c:10012.
Hiromi Suyama, Some new factors for numbers of the form 2n±1, 82T-10–230 Abstracts Amer. Math. Soc., 3(1982) 257; IV, 5 (1984) 471.
Hiromi Suyama, The cofactor of F15 is composite, 84T-10–299 Abstracts Amer. Math. Soc., 5 (1984) 271.
Paul M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comput., 64(1995) 869–888; II J. Théorie des Nombres de Bordeaux, 8(1996) 251–274; III Math. Proc. Cambridge Philos. Soc., 123(1998) 407–419; MR 95f:11022, 98h: 11037, 99b: 1 1027.
Samuel S. Wagstaff, Divisors of Mersenne numbers, Math. Comput., 40(1983) 385–397; MR 84j:10052.
Samuel S. Wagstaff, The period of the Bell exponential integers modulo a prime, in Math. Comput. 1943–1993 (Vancouver, 1993), Proc. Sympos. Appl. Math., 48(1994) 595–598; MR 95m: 1 1030.
H. C. Williams, The primality of certain integers of the form 2Arn - 1, Acta Arith., 39(1981) 7–17; MR 84h: 10012.
H. C. Williams, How was F6 factored? Math. Comput., 61 (1993) 463–474.
Samuel Yates, Titanic primes, J. Recreational Math.,16(1983–84) 265–267.
Samuel Yates, Sinkers of the Titanics, J. Recreational Math.,17(1984–85) 268–274.
Samuel Yates, Tracking Titanics, in The Lighter Side of Mathematics, Proc. Strens Mem. Conf., Calgary 1986, Math. Assoc. of America, Washington DC, Spectrum series, 1993, 349–356.
Jeff Young, Large primes and Fermat factors, Math. Comput., 67(1998) 17351738; MR 99a: 1 1010.
Jeff Young and Duncan Buell, The twentieth Fermat number is composite, Math. Comput., 50 (1988) 261–263.
Eric Bach and Jonathan Sorenson, Explicit bounds for primes in residue classes, in Math. Comput. 1943–1993 (Vancouver, 1993), Proc. Sympos. Appl. Math., (1994) -.
Carter Bays and Richard H. Hudson, The appearance of tens of billions of integers x with 7124,13 (x) 7124,1(x) in the vicinity of 1012, J. reine angew. Math.,299/300(1978) 234–237; MR 57 #12418.
Carter Bays and Richard H. Hudson, Details of the first region of integers x with 713,2(x) 713,1(x), Math. Comput.,32(1978) 571–576; MR 57 #16175.
Carter Bays and Richard H. Hudson, Numerical and graphical description of all axis crossing regions for the moduli 4 and 8 which occur before 1012, Internat. J. Math. Sci., 2(1979) 111–119; MR 80h: 10003.
Carter Bays, Kevin Ford, Richard H. Hudson and Michael Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev’s bias, J. Number Theory, 87(2001) 54–76; MR 2001m: 11148.
P. L. Chebyshev, Lettre de M. le Professeur Tschébychev à M. Fuss sur un nouveaux Théorème relatif aux nombres premiers contenus dans les formes 4n+1 et 4n + 3, Bull. Classe Phys. Acad. Imp. Sci. St. Petersburg, 11 (1853) 208.
Chen Jing-Run, On the least prime in an arithmetical progression, Sci. Sinica,14(1965) 1868–1871; MR 32 #5611.
Chen Jing-Run, On the least prime in an arithmetical progression and two theorems concerning the zeros of Dirichlet’s L-functions, Sci. Sinica,20(1977) 529–562; MR 57 #16227.
Chen Jing-Run, On the least prime in an arithmetical progression and theorems concerning the zeros of Dirichlet’s L-functions IT, Sci. Sinica, 22(1979) 859–889; MR 80k: 10042.
Chen Jing-Run and Liu Jian-Min, On the least prime in an arithmetic progression III, IV, Sci. China Ser. A,32(1989) 654–673, 792–807; MR 91h:11090ab.
Andrey Feuerverger and Greg Martin, Biases in the Shanks-Rényi prime number race, Experiment. Math.,9(2000) 535–570; 2002d:11111.
K. Ford and R. H. Hudson, Sign changes in 7rq,a(x) — 7 9,b(x), Acta Arith., 100(2001) 297–314; MR 2003a: 11121.
Kevin Ford and Sergei Konyagin, The prime number race and zeros of L-functions off the critical line, Duke Math. J., 113(2002) 313–330; MR 2003i: 11135.
S. Graham, On Linnik’s constant, Acta Arith., 39(1981) 163–179; MR 83d: 10050.
Andrew Granville and Carl Pomerance, On the least prime in certain arithmetic progressions, J. London Math. Soc.(2), 41(1990) 193–200; MR 91i: 1 1119.
D. R. Heath-Brown, Siegel zeros and the least prime in an arithmetic progression, Quart. J. Math. Oxford Ser. (2), 41(1990) 405–418; MR 91m: 1 1073.
D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions and the least prime in an arithmetic progression, Proc. London Math, Soc.(3), 64(1992) 265338; MR 93a: 1 1075.
Richard H. Hudson, A common combinatorial principle underlies Riemann’s formula, the Chebyshev phenomenon, and other subtle effects in comparative prime number theory I, J. reine angew. Math., 313 (1980) 133–150.
Richard H. Hudson, Averaging effects on irregularities in the distribution of primes in arithmetic progressions, Math. Comput., 44(1985) 561–571; MR 86h: 1 1064.
Richard H. Hudson and Alfred Brauer, On the exact number of primes in the arithmetic progressions 4n ± 1 and 6n ± 1, J. reine angew. Math.,291(1977) 23–29; MR 56 #283.
Matti Jutila, A new estimate for Linnik’s constant, Ann. Acad. Sci. Fenn. Ser. A I No. 471 (1970), 8pp.; MR 42 #5939.
Matti Jutila, On Linnik’s constant, Math. Scand., 41(1977) 45–62; MR 57 #16230.
Jerzy Kaczorowski, A contribution to the Shanks-Rényi race problem, Quart. J. Math. Oxford Ser. (2) 44(1993) 451–458; MR 94m: 1 1105.
Jerzy Kaczorowski, On the Shanks—Rényi race problem mod 5, J. Number Theory 50(1995) 106–118; MR 94m: 1 1105.
Jerzy Kaczorowski, On the distribution of primes (mod 4), Analysis, 15(1995) 159–171; MR 96h:11095.
Jerzy Kaczorowski, On the Shanks—Rényi race problem, Acta Arith., 74(1996) 31–46; MR 96k:11113.
S. Knapowski and P. Turân, Über einige Fragen der vergleichenden Primzahltheorie, Number Theory and Analysis, Plenum Press, New York, 1969, 157–171.
S. Knapowski and P. Turân, On prime numbers - 1 resp. 3 mod 4, Number Theory and Algebra,Academic Press, New York, 1977, 157–165; MR 57 #5926.
John Leech, Note on the distribution of prime numbers, J. London Math. Soc., 32 (1957) 56–58.
U. V. Linnik, On the least prime in an arithmetic progression I. The basic theorem. II. The Deuring-Heilbronn phenomenon. Rec. Math. [Mat. Sbornik] N.S.,15(57)(1944) 139–178, 347–368; MR 6, 260bc.
J. E. Littlewood, Distribution des nombres premiers, C.R. Acad. Sci. Paris, 158 (1914) 1869–1872.
Greg Martin, Asymmetries in the Shanks-Rényi prime number race, Number theory for the millenium, II (Urbana IL, 2000) 403–415, AKPeters, Natick MA,2002; MR 2004b: 11134.
Pieter Moree, Chebyshev’s bias for composite numbers, Math. Comput., 73 (2004) 425–449.
Pan Cheng-Tung, On the least prime in an arithmetic progression, Sci. Record (N.S.) 1(1957) 311–313; MR 21 #4140.
Carl Pomerance, A note on the least prime in an arithmetic progression, J. Number Theory, 12(1980) 218–223; MR 81m: 10081.
K. Prachar, Über die kleinste Primzahl einer arithmetischen Reihe, J. reine angew. Math.,206(1961) 3–4; MR 23 #A2399; and see Andrzej Schinzel, Remark on the paper of K. Prachar, 210(1962) 121–122; MR 27 #118.
Imre Z. Ruzsa, Consecutive primes modulo 4, Indag. Math. (N.S.), 12 (2001) 489–503.
Daniel Shanks, Quadratic residues and the distribution of primes, Math. Tables Aids Comput., 13 (1959) 272–284.
Wang Wei3, On the least prime in an arithmetic progression, Acta Math. Sinica(N.S.), 7(1991) 279–289; MR 93c: 1 1073.
P. D. T. A. Elliott and H. Halberstam, The least prime in an arithmetic progression, in Studies in Pure Mathematics (Presented to Richard Rado),Academic Press, London, 1971, 59–61; MR 42 #7609.
Christian Elsholtz, Triples of primes in arithmetic progressions, Q. J. Math., 53 (2002) 393–395.
P. Erdös and P. Turân, On certain sequences of integers, J. London Math. Soc., 11 (1936) 261–264.
John B. Friedlander and Henryk Iwaniec, Exceptional characters and prime numbers in arithmetic progressions, Int. Math. Res. Not., 2003 no. 37, 2033–2050.
J. Gerver, The sum of the reciprocals of a set of integers with no arithmetic progression of k terms, Proc. Amer. Math. Soc., 62 (1977) 211–214.
Joseph L. Gerver and L. Thomas Ramsey, Sets of integers with no long arithmetic progressions generated by the greedy algorithm, Math. Comput., 33 (1979) 1353–1359.
V. A. Golubev, Faktorisation der Zahlen der Form x3 ± 4x2 + 3x ± 1, Anz. Oesterreich. Akad. Wiss. Math.-Naturwiss. Kl., 1969 184–191 (see also 191–194; 297–301; 1970, 106–112; 1972, 19–20, 178–179 ).
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progression, April 2004 preprint.
Emil Grosswald, Long arithmetic progressions that consist only of primes and almost primes, Notices Amer. Math. Soc., 26 (1979) A451.
Emil Grosswald, Arithmetic progressions of arbitrary length and consisting only of primes and almost primes, J. reine angew. Math., 317 (1980) 200–208.
Emil Grosswald, Arithmetic progressions that consist only of primes, J. Number Theory, 14 (1982) 9–31.
Emil Grosswald and Peter Hagis, Arithmetic progressions consisting only of primes, Math. Comput., 33(1979) 1343–1352; MR 80k:10054.
H. Halberstam, D. R. Heath-Brown and H.-E. Richert, On almost-primes in short intervals, in Recent Progress in Analytic Number Theory, Vol. 1, Academic Press, 1981, 69–101; MR 83a: 10075.
D. R. Heath-Brown, Almost-primes in arithmetic progressions and short intervals, Math. Proc. Cambridge Philos. Soc.,83(1978) 357–375; MR 58 #10789.
D. R. Heath-Brown, Three primes and an almost-prime in arithmetic progression, J. London Math. Soc., (2) 23 (1981) 396–414.
D. R. Heath-Brown, Siegel zeros and the least prime in an arithmetic progression, Quart. J. Math. Oxford Ser. (2), 41(1990) 405–418; MR 91m: 1 1073.
D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc.(3), 64(1992) 265338; MR 93a: 1 1075.
Edgar Karst, 12–16 primes in arithmetical progression, J. Recreational Math., 2 (1969) 214–215.
Edgar Karst, Lists of ten or more primes in arithmetical progression, Scripta Math., 28 (1970) 313–317.
Edgar Karst and S. C. Root, Teilfolgen von Primzahlen in arithmetischer Progression, Anz. Oesterreich. Akad. Wiss. Math.-Naturwiss. KI.,1972, 19–20 (see also 178–179).
A. Kumchev, The difference between consecutive primes in an arithmetic progression, Q. J. Math., 53 (2002) 479–501.
U. V. Linnik, On the least prime in an arithmetic progression, I. The basic theorem, Rec. Math. [Mat. Sbornik] N.S.,15(57)(1944) 139–178; II. The Deuring-Heilbronn phenomenon, 347–368; MR 6, 260bc.
Carl Pomerance, The prime number graph, Math. Comput., 33(1979) 399408; MR 80d: 10013.
Paul A. Pritchard, Andrew Moran and Anthony Thyssen, Twenty-two primes in arithmetic progression, Math. Comput., 64(1995) 1337–1339; MR 95j: 1 1003.
Olivier Ramaré and Robert Rumely, Primes in arithmetic progressions, Math. Comput.,65(1996) 397–425; MR 97a:11144.
W. Sierpinski, Remarque sur les progressions arithmétiques, Colloq. Math., 3 (1955) 44–49.
Sol Weintraub, Primes in arithmetic progression, BIT 17 (1977) 239–243.
K. Zarankiewicz, Problem 117, Colloq. Math.,3(1955) 46, 73.
S. Chowla, There exists an infinity of 3-combinations of primes in A.P., Proc. Lahore Philos. Soc., 6 no. 2(1944) 15–16; MR 7, 243.
J. G. van der Corput, Über Summen von Primzahlen und Primzahlquadraten, Math. Annalen, 116(1939) 1–50; Zbl. 19, 196.
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson and P. Zimmermann, Ten consecutive primes in arithmetic progression. Math. Comput. 71(2002) 1323–1328; MR 2003d: 11137.
H. Dubner and H. Nelson, Seven consecutive primes in arithmetic progression, Math. Comput., 66(1997) 1743–1749; MR 98a: 1 1122.
P. Erdös and A. Rényi, Some problems and results on consecutive primes, Simon Stevin, 27(1950) 115–125; MR 11, 644.
M. F. Jones, M. Lal and W. J. Blundon, Statistics on certain large primes, Math. Comput.,21(1967) 103–107; MR 36 #3707.
L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comput., 21 (1967) 489.
H. L. Nelson, There is a better sequence, J. Recreational Math., 8 (1975) 39–43.
S. Weintraub, Consectutive primes in arithmetic progression, J. Recreational Math., 25 (1993) 169–171.
Takashi Agoh, On Sophie Germain primes, Tatra Mt. Math. Publ., 20(2000) 65–73; MR 2002d: 11008.
Harvey Dubner, Large Sophie Germain primes, Math. Comput., 65(1996) 393–396; MR 96d: 1 1008.
Claude Lalout and Jean Meeus, Nearly-doubled primes, J. Recreational Math.,13 (1980/81) 30–35.
D. H. Lehmer, Tests for primality by the converse of Fermat’s theorem, Bull. Amer. Math. Soc., 33 (1927) 327–340.
D. H. Lehmer, On certain chains of primes, Proc. London Math. Soc., 14A (Littlewood 80 volume, 1965 ) 183–186.
Günter Löh, Long chains of nearly doubled primes, Math. Comput., 53(1989) 751–759; MR 90e: 1 1015.
Edlyn Teske and Hugh C. Williams, A note on Shanks’s chains of primes, Algorithmic number theory (Leiden, 2000), Springer Lecture Notes in Comput. Sci. 1838(2000) 563–580; MR 2002k: 11228.
Danilo Bazzanella, Primes in almost all short intervals, Boll. Un. Mat. Ital.(7), 9(1995) 233–249; MR99c:11089.
Richard Blecksmith, Paul Erdôs and J. L. Selfridge, Cluster primes, Amer. Math. Monthly, 106(1999) 43–48; MR 2000a: 11126.
E. Bombieri and H. Davenport, Small differences between prime numbers, Proc. Roy. Soc. Ser. A, 293(1966)1–18; MR 33 #7314.
Richard P. Brent, The first occurrence of large gaps between successive primes, Math. Comput., 27 (1973) 959–963; MR 48 #8360; (and see Math. Comput., 35 (1980) 1435–1436.
Richard P. Brent, Irregularities in the distribution of primes and twin primes, Math. Comput., 29 (1975) 43–56.
V. Brun, Le crible d’Eratosthène et le théorème de Goldbach, C. R. Acad. Sci. Paris, 168 (1919) 544–546.
J. H. Cadwell, Large intervals between consecutive primes, Math. Comput., 25 (1971) 909–913.
Cai Ying-Chun and Lu Ming-Gao, On the upper bound for 7r2(x), Acta Arith., 110 (2003) 275–298.
Chen Jing-Run, On the distribution of almost primes in an interval II, Sci. Sinica, 22(1979) 253–275; Zbl., 408.10030.
Chen Jing-Run and Wang Tian-Ze, Acta Math. Sinica, 32(1989) 712–718; MR 91e:11108.
Chen Jing-Run and Wang Tian-Ze, On distribution of primes in an arithmetical progression, Sci. China Ser. A, 33(1990) 397–408; MR 91k: 1 1078.
H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith., 2 (1937) 23–46.
Pamela A. Cutter, Finding prime pairs with particular gaps, Math. Comput., 70(2001) 1737–1744 (electronic); MR 2002c: 11174.
TamAs Dénes, Estimation of the number of twin primes by application of the complementary prime sieve, Pure Math. Appl., 13(2002) 325–331; MR 2003m: 11153.
Christian Elsholtz, On cluster primes, Acta Arith., 109(2003) 281–284; MR 2004c:11169.
Paul Erdös and Melvyn B. Nathanson, On the sums of the reciprocals of the differences between consecutive primes, Number Theory (New York, 1991–1995, Springer, New York, 1996, 97–101; MR 97h: 1 1094.
Anthony D. Forbes, A large pair of twin primes, Math. Comput., 66(1997)451455; MR 99c: 1 1111.
Etienne Fouvry, in Analytic Number Theory (Tokyo 1988) 65–85, Lecture Notes in Math., 1434, Springer, Berlin, 1990; MR 91g:11104. É.
Fouvry and F. Grupp, On the switching principle in sieve theory, J. reine angew. Math., 370(1986) 101–126; MR 87j: 1 1092.
J. B. Friedlander and J. C. Lagarias, On the distribution in short intervals of integers having no large prime factor, J. Number Theory, 25 (1987) 249–273.
J. W. L. Glaisher, On long successions of composite numbers, Messenger of Math., 7(1877) 102–106, 171–176.
D. A. Goldston, On Bombieri and Davenport’s theorem concerning small gaps between primes, Mathematika, 39(1992) 10–17; MR 93h: 1 1102.
Glyn Harman, Short intervals containing numbers without large prime factors, Math. Proc. Cambridge Philos. Soc., 109(1991) 1–5; MR 91h: 1 1093.
Jan Kristian Haugland, Large prime-free intervals by elementary methods, Normat, 39 (1991) 76–77.
D. R. Heath-Brown, The differences between consecutive primes III, J. London Math. Soc.(2) 20(1979) 177–178; MR 81f:10055.
Y. K. Huen, The twin prime problem revisited, Internat. J. Math. Ed. Sci. Tech., 28 (1997) 825–834.
Martin Huxley, An application of the Fouvey-Iwaniec theorem, Acta Arith., 43 (1984) 441–443.
Karl-Heinz Indlekofer and Antal Jârai, Largest known twin primes, Math. Comput., 65(1996) 427–428; MR 96d: 1 1009.
Karl-Heinz Indlekofer and Antal Jârai, Largest known twin primes and Sophie Germain primes, Math. Comput., 68(1999) 1317–1324; MR 99k: 1 1013.
H. Iwaniec and M. Laborde, P2 in short intervals, Ann. Inst. Fourier(Grenoble), 31(1981) 37–56; MR 83e: 10061.
Jia Chao-Hua, Three primes theorem in a short interval VI (Chinese), Acta Math. Sinica, 34(1991) 832–850; MR 93h: 1 1104.
Jia Chao-Hua, Difference between consecutive primes, Sci. China Ser. A, 38(1995) 1163–1186; MR 99m: 1 1081.
Jia Chao-Hua, Almost all short intervals containing prime numbers, Acta Arith., 76(1996) 21–84; MR 97e:11110.
Li Hong-Ze, Almost primes in short intervals, Sci. China Ser. A, 37(1994) 1428–1441; MR 96e: 1 1116.
Li Hong-Ze, Primes in short intervals, Math. Proc. Cambridge Philos. Soc., 122(1997) 193–205; MR 98e: 1 1103.
Liu Hong-Quan, On the prime twins problem, Sci. China Ser. A, 33(1990) 281–298; MR 91i: 1 1125.
Liu Hong-Quan, Almost primes in short intervals, J. Number Theory, 57(1996) 303–302; MR 97c: 1 1092.
Lou Shi-Tuo and Qi Yao, Upper bounds for primes in intervals (Chinese), Chinese Ann. Math. Ser. A, 10(1989) 255–262; MR 91d: 1 1112.
Helmut Maier, Small differences between prime numbers, Michigan Math. J., 35 (1988) 323–344.
Helmut Maier, Primes in short intervals, Michigan Math. J., 32 (1985) 221–225.
Helmut Maier and Carl Pomerance, Unusually large gaps between consecutive primes, Théorie des nombres, (Quebec, PQ, 1987), de Gruyter, 1989, 625–632; MR 91a:11045: and see Trans. Amer. Math. Soc., 322(1990) 201–237; MR 91b: 1 1093.
H. Maier and Cam Stewart, On intervals with few prime numbers, (2003 preprint)
Hiroshi Mikawa, On prime twins in arithmetic progressions, Tsukuba J. Math., 16(1992) 377–387; MR 94e: 1 1101.
C. J. Mozzochi, On the difference between consecutive primes, J. Number Theory 24 (1986) 181–187.
Thomas R. Nicely, Enumeration to 1014 of the twin primes and Brun’s constant, Virginia J. Sci., 46(1995) 195–204; reviewed by Richard P. Brent, Math. Comput., 66(1997) 924–925; MR 97e: 1 1014.
Thomas R. Nicely, New maximal prime gaps and first occurrences, Math. Comput., 68(1999) 1311–1315; MR 99i: 1 1004.
Thomas R. Nicely, A new error analysis for Brun’s constant, Virginia J. Sci., 52(2001) 45–55; MR 2003d: 11184.
Bertil Nyman and Thomas R. Nicely, New prime gaps between 1015 and 5x 1016, J. Integer Seq. 6(2003) Article 03.3.1, 6 pp. (electronic).
Bodo K. Parady, Joel F. Smith and Sergio E. Zarantonello, Largest known twin primes, Math. Comput., 55(1990) 381–382; MR 90j: 1 1013.
A. S. Peck, Differences between consecutive primes, Proc. London Math. Soc., (3) 76(1998) 33–69; MR 98i: 1 1071.
G. Z. Pil’tyai, The magnitude of the difference between consecutive primes (Russian), Studies in number theory 4, Izdat. Saratov. Univ., Saratov, 1972, 73–79; MR 52 #13680.
Janos Pintz, Very large gaps between consecutive primes, J. Number Theory, 63(1997) 286–301; MR 98c: 1 1092.
Olivier Ramaré Si Yannick Saouter, Short effective intervals containing primes, J. Number Theory 98(2003) 10–33; MR 2004a:11095.
S. Salerno and A. Vitolo, p+ P2 in short intervals, Note Mat., 13(1993) 309–328; MR 96c: 1 1109.
Jszsef Sândor, Despre §irusi, serü si aplicatü in teoria numerelor prime (On sequences, series and applications in prime number theory. Romanian), Gaz. Mat. Perf. Met., 6 (1985) 38–48.
Daniel Shanks, On maximal gaps between successive primes, Math. Comput., 18(1964) 646–651; MR 29 #4745.
Daniel Shanks and John W. Wrench, Brun’s constant, Math. Comput., 28(1974) 293–299; corrigenda, ibid. 28(1974) 1183; MR 50 #4510.
S. Uchiyama, On the difference between consecutive prime numbers, Acta Arith., 27 (1975) 153–157.
Nigel Watt, Short intervals almost all containing primes, Acta Arith., 72(1995) 131–167; MR 96f:11115.
Wu Jiel, Sur la suite des nombres premiers jumeaux, Acta Arith. 55(1990) 365–394; MR 91j:11074.
Wu Jiel, Thôrmes généralisés de Bombieri-Vinogradov dans les petits intervalles, Quart. J. Math. Oxford Ser.(2), 44(1993) 109–128; MR 93m:11090.
Jeff Young and Aaron Potler, First occurrence prime gaps, Math. Comput., 52 (1989) 221–224.
Yu Gang’, The differences between consecutive primes, Bull. London Math. Soc., 28(1996) 242–248; MR 97a: 1 1142.
Alessandro Zaccagnini, A note on large gaps between consecutive primes in arithmetic progressions, J. Number Theory, 42 (1992) 100–102.
Antal Balog, The prime k-tuplets conjecture on the average, Analytic Number Theory (Allerton Park, IL, 1989 ), 47–75, Progr. Math., 85, Birkhäuser, Boston, 1990.
Pierre Dusart, Sur la conjecture ir(x+y) x(x)-F-rr(y), Acta Arith., 102 (2002) 295–308.
Paul Erdös and Ian Richards, Density functions for prime and relatively prime numbers, Monatsh. Math. 83(1977) 99–112; Zbl. 355.10034.
Anthony D. Forbes, Prime clusters and Cunningham chains, Mathy. Comput., 68(1999) 1739–1747; MR 99m: 1 1007.
Anthony D. Forbes, Large prime triplets, Math. Spectrum 29(1996–97) 65.
Anthony D. Forbes, Prime k-tuplets–15, M500, 156(July 1997 ) 14–15.
John B. Friedlander and Andrew Granville, Limitations to the equi-distribution of primes I, IV, III, Ann. of Math.(2), 129(1989) 363–382; MR 90e:11125; Proc. Roy. Soc. London Ser. A, 435(1991) 197–204; MR 93g:11098; Compositio Math., 81 (1992) 19–32.
John B. Friedlander, Andrew Granville, Adolf Hildebrandt and Helmut Maier, Oscillation theorems for primes in arithmetic progressions and for sifting functions, J. Amer. Math. Soc., 4(1991) 25–86; MR 92a: 1 1103.
R. Garunkstis, On some inequalities concerning 7(x), Experiment. Math., 11(2002) 297–301; MR 2003k: 11143.
D. R. Heath-Brown, Almost-prime k-tuples, Mathematika, 44 (1997) 245–266.
Douglas Hensley and Ian Richards, On the incompatibility of two conjectures concerning primes, Proc. Symp. Pure Math., (Analytic Number Theory, St. Louis, 1972 ) 24 123–127.
H. Iwaniec, On the Brun-Titchmarsh theorem, J. Math. Soc. Japan, 34(1982) 95–123; MR 83a: 10082.
John Leech, Groups of primes having maximum density, Math. Tables Aids Comput. 12(1958) 144–145; MR 20 #5163.
H. L. Montgomery and R. C. Vaughan, The large sieve, Mathematika 20(1973) 119–134; MR 51 #10260.
Laurentiu Panaitopol, On the inequality papy pay, Bull. Math. Soc. Sci. Math. Roumanie (N.S., 41 (89) (1998) 135–139.
Laurentiu Panaitopol, An inequality related to the Hardy-Littlewood conjecture, An. Univ. Bucuresti Mat., 49(2000) 163–166; MR 2003c: 11114.
Laurentiu Panaitopol, Checking the Hardy-Littlewood conjecture in special cases, Rev. Roumaine Math. Pures Appl., 46(2001) 465–470; MR 2003e: 11098.
Ian Richards, On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem, Bull. Amer. Math. Soc., 80 (1974) 419–438.
Herschel F. Smith, On a generalization of the prime pair problem, Math. Tables Aids Comput. 11(1957) 249–254; MR 20 #833.
Warut Roonguthai, Large prime quadruplets, M500 153 (Dec.1996) 4–5.
Charles W. Trigg, A large prime quadruplet, J. Recreational Math. 14 (1981/82) 167.
Sheng-Gang Xie, The prime 4-tuplet problem (Chinese. English summary), Sichuan Daxue Xuebao, 26(1989) 168–171; MR 91f: 1 1066.
R. B. Killgrove and K. E. Ralston, On a conjecture concerning the primes, Math. Tables Aids Comput. 13(1959) 121–122; MR 21 #4943.
Andrew M. Odlyzko, Iterated absolute values of differences of consecutive primes, Math. Comput., 61(1993) 373–380; MR 93k: 1 1119.
F. Proth, Sur la série des nombres premiers, Nouv. Corresp. Math., 4 (1878) 236–240.
P. Eras, On the difference of consecutive primes, Bull. Amer. Math. Soc., 54(1948) 885–889; MR 10, 235.
P. Eras and P. Turân, On some new questions on the distribution ofprime numbers, Bull. Amer. Math. Soc., 54(1948) 371–378; MR 9, 498.
Laurentiu Panaitopol, Erdös-Turân problem for prime numbers in arithmetic progression, Math. Rep, (Bucur.), 1(51)(1999) 595–599; MR 2002d: 11112.
William Adams and Daniel Shanks, Strong primality tests that are not sufficient, Math. Comput., 39 (1982) 255–300.
Richard André-Jeannin, On the existence of even Fibonacci pseudoprimes with parameters P and Q, Fibonacci Quart., 34(1996) 75–78; MR 96m: 1 1013.
F. Arnault, The Rabin-Monier theorem for Lucas pseudoprimes, Math. Comput., 66(1997) 869–881; MR 97f: 1 1009.
Steven Arno, A note on Perrin pseudoprimes, Math. Comput., 56(1991) 371376. 371–376; MR 91k: 1 1011.
N. G. W. H. Beeger, On even numbers m dividing 2’ — 2, Amer. Math. Monthly, 58(1951) 553–555; MR 13, 320.
Jerzy Browkin, Some new kinds of pseudoprimes, Math. Comput., 73 (2004) 1031–1037.
Paul S. Bruckman, On the infinitude of Lucas pseudoprimes, Fibonacci Quart., 32 (1994) 153–154; MR 95c: 1 1011.
Paul S. Bruckman, Lucas pseudoprimes are odd, Fibonacci Quart., 32 (1994) 155–157; MR 95c: 1 1012.
Paul S. Bruckman, On a conjecture of Di Porto and Filipponi, Fibonacci Quart., 32(1994) 158–159; MR 95c: 1 1013.
Paul S. Bruckman, Some interesting subsequences of the Fibonacci and Lucas pseudoprimes, Fibonacci Quart., 34(1996) 332–341; MR 97e: 1 1022.
Bui Minh-Phong, On the distribution of Lucas and Lehmer pseudoprimes. Festscrift for the 50th birthday of Karl-Heinz Indlekofer, Ann. Univ. Sci. Budapest. Sect. Comput., 14(1994) 145–163; MR 96d: 1 1106.
Walter Carlip, Eliot Jacobson and Lawrence Somer, Pseudoprimes, perfect numbers and a problem of Lehmer, Fibonacci Quart., 36(1998) 361–371; MR 99g: 1 1013.
R. D. Carmichael, On composite numbers P which satisfy the Fermat congruence a1’-1 w 1 mod P, Amer. Math. Monthly, 19 (1912) 22–27.
M. Cipolla, Sui numeri composti P the verificiano la congruenza di Fermat, aP-1 w 1 (mod P), Annali di Matematica, 9 (1904) 139–160.
John H. Conway, Richard K. Guy, William A. Schneeberger and N. J. A. Sloane, The primary pretenders, Acta. Arith., 78(1997) 307–313; MR 197k: 1 1004.
Adina Di Porto, Nonexistence of even Fibonacci pseudoprimes of the 1st kind, Fibonacci Quart., 31(1993) 173–177; MR94d:11007.
L. A. G. Dresel, On pseudoprimes related to generalized Lucas sequences, Fibonacci Quart., 35(1997) 35–42; MR 98b: 1 1010.
P. Erdös, On the converse of Fermat’s theorem, Amer. Math. Monthly, 56 (1949) 623–624; MR 11, 331.
P. Erdös, On almost primes, Amer. Math. Monthly, 57(1950) 404–407; MR 12, 80.
P. Erdös, P. Kiss and A. Sârközy, A lower bound for the counting function of Lucas pseudoprimes, Math. Comput., 51(1988) 259–279; MR 89e: 1 1011.
P. Erdös and C. Pomerance, The number of false witnesses for a composite number, Math. Comput., 46(1986) 259–279; MR 87i: 1 1183.
J. B. Friedlander, Shifted primes without large prime factors, in Number Theory and Applications, (Proc. NATO Adv. Study Inst., Banff, 1988 ), Kluwer, Dordrecht, 1989, 393–401;
Horst W. Gerlach, On liars and witnesses in the strong pseudoprimality test, New Zealand J. Math., 24 (1995) 5–15.
Daniel M. Gordon and Carl Pomerance, The distribution of Lucas and elliptic pseudoprimes, Math. Comput., 57(1991) 825–838; MR 92h:11081; corrigendum 60(1993) 877; MR 93h: 1 1108.
S. Gurak, Pseudoprimes for higher-order linear recurrence sequences, Math. Comput., 55 (1990) 783–813.
S. Gurak, On higher-order pseudoprimes of Lehmer type, Number Theory (Halifax NS, 1994) 215–227, CMS Conf. Proc., 15, Amer. Math. Soc., 1995; MR 96f: 1 1167.
Hideji Ito, On elliptic pseudoprimes, Mem. College Ed. Akita Univ. Natur. Sci., 46(1994) 1–7; MR 95e:11009.
Gerhard Jaeschke, On strong pseudoprimes to several bases, Math. Comput., 61(1993) 915–926; MR 94d: 1 1004.
I. Joô, On generalized Lucas pseudoprimes, Acta Math. Hungar., 55 (1990) 315–322.
I. Joô and Phong Bui-Minh, On super Lehmer pseudoprimes, Studia Sci. Math. Hungar., 25(1990) 121–124; MR 92d: 1 1109.
Kim Su-Hee and Carl Pomerance, The probability that a random probable prime is composite, Math. Comput., 53(1989) 721–741; MR 90e: 1 1190.
Péter Kiss and Phong Bui-Minh, On a problem of A. Rotkiewicz, Math. Comput., 48(1987) 751–755; MR 88d: 1 1004.
G. Kowol, On strong Dickson pseupoprimes, Appl. Algebra Engrg. Comm. Comput., 3(1992) 129–138; MR 96g: 1 1005.
D. H. Lehmer, On the converse of Fermat’s theorem, Amer. Math. Monthly, 43(1936) 347–354; II 56(1949) 300–309; MR 10, 681.
Rudolf Lidl, Winfried B. Müller and Alan Oswald, Some remarks on strong Fibonacci pseudoprimes, Appl. Algebra Engrg. Comm. Comput., 1(1990) 59–65; MR 95k: 1 1015.
Andrzej Makowski, On a problem of Rotkiewicz on pseudoprime numbers, Elem. Math., 29 (1974) 13.
A. Makowski and A. Rotkiewicz, On pseudoprime numbers of special form, Colloq. Math., 20(1969) 269–271; MR 39 #5458.
Wayne L. McDaniel, Some pseudoprimes and related numbers having special forms, Math. Comput., 53(1989) 407–409; MR 89m: 1 1006.
F. Morain, Pseudoprimes: a survey of recent results, Eurocode ’82, Springer Lecture Notes, 339(1993) 207–215; MR 95c: 1 1172.
Siguna Müller, On strong Lucas pseudoprimes, Contributions to general algebra, 10 (Klagenfurt, 1997) 237–249, Heyn, Klagenfurt, 1998; MR 99k: 1 1008.
Siguna Müller, A note on strong Dickson pseudoprimes, Appl. Algebra Engrg. Comm. Comput., 9(1998) 247–264; MR 99j: 1 1008.
Winfried B. Müller and Alan Oswald, Generalized Fibonacci pseudoprimes and probablr primes, Applications of Fibonacci numbers, Vol.5 (St. Andrews, 1992), 459–464, Kluwer Acad. Publ.. Dordrecht, 1993; MR 95f: 1 1105.
R. Perrin, Item 1484, L’Intermédiaire des mathématicians, 6(1899) 76–77; see also E. Malo, ibid., 7(1900) 312 and E. B. Escott, ibid., 8 (1901) 63–64.
Richard G. E. Pinch, The pseudoprimes up to 1013, Algorithmic Number Theory (Leiden, 2000), Springer Lecture Notes in Comput. Sci., 1838(2000) 459–473; MR 2002g: 11177.
Carl Pomerance, A new lower bound for the pseudoprime counting function, Illinois J. Math., 26(1982) 4–9; MR 83h:10012.
Carl Pomerance, On the distribution of pseudoprimes, Math. Comput., 37 (1981) 587–593; MR 83k:10009.
Carl Pomerance, Two methods in elementary analytic number theory, in Number Theory and Applications, (Proc. NATO Adv. Study Inst., Banff, 1988), Kluwer, Dordrecht, 1989, 135–161;
Carl Pomerance, John L. Selfridge and Samuel S. Wagstaff, The pseudoprimes to 25. 109, Math. Comput., 35(1980) 1003–1026; MR 82g:10030.
A. Rotkiewicz, Pseudoprime Numbers and their Generalizations, Student Association of the Faculty of Sciences, Univ. of Novi Sad, 1972; MR 48 #8373; Zbl. 324. 10007.
A. Rotkiewicz, Sur les diviseurs composés des nombres an — b’, Bull. Soc. Roy. Sci. Liège, 32(1963) 191–195; MR 26 #3645.
A. Rotkiewicz, Sur les nombres pseudopremiers de la forme ax + b, Comptes Rendus Acad. Sci. Paris, 257(1963) 2601–2604; MR 29 #61.
A. Rotkiewicz, Sur les formules donnant des nombres pseudopremiers, Colloq. Math. 12(1964) 69–72; MR 29 #3416.
A. Rotkiewicz, Sur les nombres pseudopremiers de la forme nk + 1, Elern. Math., 21(1966) 32–33; MR 33 #112.
A. Rotkiewicz, On Euler-Lehmer pseudoprimes and strong Lehmer pseudo-primes with parameters L, Q in arithmetic progressions, Math. Comput., 39(1982) 239–247; MR 83k:10004.
A. Rotkiewicz, On the congruence 2n-2–1(mod n), Math. Comput., 43 (1984) 271–272; MR 85e: 1 1005.
A. Rotkiewicz, On strong Lehmer pseudoprimes in the case of negative discriminant in arithmetic progressions, Acta Arith., 68(1994) 145–151; MR 96h: 1 1008.
A. Rotkiewicz, Arithmetical progressions formed by k different Lehmer pseudoprimes, Rend. Circ. Mat. Palermo (2), 43(1994) 391–402; MR 96f: 1 1026.
A. Rotkiewicz, On Lucas pseudoprimes of the form ax2 +bxy + cy2, Applications of Fibonacci numbers, Vol. 6 (Pullman WA 1994), 409–421, Kluwer Acad. Publ., Dordrecht 1996; MR9 10: 1 1001.
A. Rotkiewicz, On Lucas pseudoprimes of the form ax2 + bxy + cy2, Applications of Fibonacci numbers, Vol. 6 (Pullman WA, 1994) 409–421, Kluwer Acad. Publ. Dordrecht, 1996; MR 97d: 1 1008.
A. Rotkiewicz, On the theorem of Wojcik, Glasgow Math. J., 38(1996) 157–162; MR 97i: 1 1010.
A. Rotkiewicz, On Lucas pseudoprimes of the form ax2 + bxy + cy2 in arithmetic progressions AX + B with a prescribed value of the Jacobi symbol, Acta Math. Inform. Univ. Ostraviensis, 10(2002) 103–109; MR 2003i: 11004.
A. Rotkiewicz and K. Ziemak, On even pseudoprimes, Fibonacci Quart., 33 (1995) 123–125; MR 96c: 1 1005.
Shen Mok-Kong, On the congruence 2n-k w 1(mod n), Math. Comput., 46(1986) 715–716; MR 87e: 1 1005.
Lawrence Somer, On even Fibonacci pseudoprimes, Applications of Fibonacci numbers, 4(1990), Kluwer, 1991, 277–288; MR 94b: 1 1014.
George Szekeres, Higher order pseudoprimes in primality testing, Combina-tories, Paul Erdfis is eighty, Vol.2 (Keszthely, 1993 ) 451–458, Bolyai Soc. Math. Stud., 2 Janos Bolyai Math. Soc., Budapest, 1996.
K. Szymiczek, On prime numbers p, q and r such that pq, pr and qr are pseudoprimes, Colloq. Math., 13(1964–65) 259–263; MR 31 #4757.
K. Szymiczek, On pseudoprime numbers which are products of distinct primes, Amer. Math. Monthly, 74(1967) 35–37; MR 34 #5746.
S. S. Wagstaff, Pseudoprimes and a generalization of Artin’s conjecture, Acta Arith., 41(1982) 151–161; MR 83m: 10004.
Masataka Yorinaga, Search for absolute pseudoprime numbers (Japanese), Siîgaku, 31 (1979) 374–376; MR 82c: 10008.
Zhang Zhen-Xiang, Finding strong pseudoprimes to several bases, Math. Comput., 70(2001) 863–872; MR 2001g: 11009.
Zhang Zhen-Xiang and Min Tang, Finding strong pseudoprimes to several bases, II, Math. Comput. 72(2003) 2085–2097; MR 2004c:11008.
W. R. Alford and Jon Grantham, There are Carmichael numbers with many prime factors, (2003 preprint)
W. Red Alford, Andrew Granville and Carl Pomerance, On the difficulty of finding reliable witnesses, in Algorithmic Number Theory (Ithaca, NY, 1994), 116, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994; MR 96d: 1 1136.
W. Red Alford, Andrew Granville and Carl Pomerance, There are infinitely many Carmichael numbers, Ann. of Math.(2), 139(1994) 703–722; MR 95k: 1 1114.
François Arnault, Constructing Carmichael numbers which are strong pseudo-primes to several bases, J. Symbolic Comput., 20(1995) 151–161; MR 96k: 1 1153.
Robert Baillie and Samuel S. Wagstaff, Lucas pseudoprimes, Math. Comput., 35 (1980) 1391–1417.
Ramachandran Balasubramanian and S. V. Nagaraj, Density of Carmichael numbers with three prime factors, Math. Comput., 66(1997) 1705–1708; MR 98d: 1 1110.
N. G. W. H. Beeger, On composite numbers n for which a„ _i = 1 mod n for every a prime to n, Scripta Math., 16 (1950) 133–135.
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc., 16(1909–10) 232–238.
Harvey Dubner, A new method for producing large Carmichael numbers, Math. Comput., 53(1989) 411–414; MR 89m: 1 1013.
Harvey Dubner, Carmichael numbers and Egyptian fractions, Math. Japon., 43(1996) 411–419; MR 97d: 1 1011.
Harvey Dubner, Carmichael numbers of the form (6m+1)(12m+1)(18m+1), J. Integer Seq., 5(2002) A02.2.1, 8pp. (electronic); MR 2003g: 11144.
H. J. A. Duparc, On Carmichael numbers, Simon Stevin, 29(1952) 21–24; MR 14, 21f.
P. Erdös, On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen, 4(1956) 201–206; MR 18, 18.
Andrew Granville, Prime testing and Carmichael numbers, Notices Amer. Math. Soc., 39 (1992) 696–700.
Andrew Granville and Carl Pomerance, Two contradictory conjectures concerning Carmichael numbers, Math. Comput., 71(2002) 883–908; MR 2003d: 11148.
D. Guillaume, Table de nombres de Carmichael inférieurs à 1012, preprint, May 1991
D. Guillaume and F. Morain, Building Carmichael numbers with a large number of prime factors and generalization to other numbers, preprint June, 1992.
Jay Roderick Hill, Large Carmichael numbers with three prime factors, Abstract 79T-A136, Notices Amer. Math. Soc., 26(1979) A-374.
Gerhard Jaeschke, The Carmichael numbers to 1012, Math. Comput., 55 (1990) 383–389; MR 90m: 1 1018.
I. Job and Phong Bui-Minh, On super Lehmer pseudoprimes, Studia Sci. Math. Hungar., 25 (1990) 121–124.
W. Keller, The Carmichael numbers to 1013, Abstracts Amer. Math. Soc., 9 (1988) 328–329.
W. Knödel, Eine obere Schranke für die Anzahl der Carmichaelschen Zahlen kleiner als x, Arch. Math., 4(1953) 282–284; MR 15, 289 (and see Math. Nachr., 9 (1953) 343–350 ).
A. Korselt, Problème chinois, L’intermédiaire des math., 6 (1899) 142–143.
D. H. Lehmer, Strong Carmichael numbers, J. Austral. Math. Soc. Ser. A, 21 (1976) 508–510.
G. Löh, Carmichael numbers with a large number of prime factors, Abstracts Amer. Math. Soc., 9(1988) 329; II (with W. Niebuhr) 10 (1989) 305.
Günter Löh and Wolfgang Niebuhr, A new algorithm for constructing large Carmichael numbers, Math. Comput., 65(1996) 823–836; MR 96g: 1 1123.
Frantisék Marko, A note on pseudoprimes with respect to abelian linear recurring sequence, Math. Slovaca, 46(1996) 173–176; MR 97k: 1 1009.
Rex Matthews, Strong pseudoprimes and generalized Carmichael numbers, Finite fields: theory, applications and algorithms (Las Vegas NV 1993), 227–223, Contemp. Math., 168 Amer. Math. Soc., 1994; MR 95g: 1 1125.
Siguna M. S. Müller, Carmichael numbers and Lucas tests, Finite fields: theory, applications and algorithms (Waterloo ON, 1997) 193–202, Contemp. Math., 225, Amer. Math. Soc., 1999; MR 99m: 1 1008.
R. G. E. Pinch, The Carmichael numbers up to 1015, Math. Comput., 61 (1993) 381–391; MR 93m: 1 1137.
A. J. van der Poorten and A. Rotkiewicz, On strong pseudoprimes in arithmetic progressions, J. Austral. Math. Soc. Ser. A, 29 (1980) 316–321.
S. S. Wagstaff, Large Carmichael numbers, Math. J. Okayama Univ., 22 (1980) 33–41; MR 82c:10007.
H. C. Williams, On numbers analogous to Carmichael numbers, Canad. Math. Bull., 20 (1977) 133–143.
Dale Woods and Joel Huenemann, Larger Carmichael numbers, Comput. Math. Appl., 8(1982) 215–216; MR 83f:10017.
Masataka Yorinaga, Numerical computation of Carmichael numbers, I, II, Math. J. Okayama Univ. 20(1978) 151–163, 21(1979) 183–205; MR 80d:10026, 80j:10002.
Masataka Yorinaga, Carmichael numbers with many prime factors, Math. J. Okayama Univ., 22(1980) 169–184; MR 81m:10018.
Zhang Ming-Zhi, A method for finding large Carmichael numbers, Sichuan Daxue Xuebao, 29(1992) 472–479; MR 93m:11009.
Zhang Zhen-Xiang, A one-parameter quadratic-base version of the Baillie PSW probable prime test, Math. Comput., 71 (2002) 1699–1734.
Zhu Wen-Yu, Carmichael numbers that are strong pseudoprimes to some bases, Sichuan Daxue Xuebao, 34 (1997) 269–275.
William W. Adams, Eric Liverance and Daniel Shanks, Infinitely many necessary and sufficient conditions for primality, Bull. Inst. Combin. Appl., 3(1991) 69–76; MR 90e: 1 1011.
Takashi Agoh, On Giuga’s conjecture, Manuscripta Math., 87(1995) 501–510; MR 96f:11005.
Takashi Agoh, Karl Dilcher and Ladislav Skula, Wilson quotients for composite moduli, Math. Comput., 67 (1998) 843–861.
Jesûs Almansa and Leonardo Prieto, New formulas for the nth prime, (spanish) Lect. Math., 15 (1994) 227–231.
A. R. Ansari, On prime representing function, Ganita, 2(1951) 81–82; MR 15, 11.
R. C. Baker, The greatest prime factor of the integers in an interval, Acta Arith., 47(1986) 193–231; MR 88a: 1 1086.
R. C. Baker and Glyn Harman, Numbers with a large prime factor, Acta Arith., 73(1995) 119–145; MR 97a: 1 1138.
Thoger Bang, A function representing all prime numbers, Norsk Mat. Tidsskr., 34(1952) 117–118; MR 14, 621.
V. I. Baranov and B. S. Stechkin, Extremal Combinatorial Problems and their Applications, Kluwer, 1993, Problem 2. 22.
Paul T. Bateman and Harold G. Diamond, A hundred years of prime numbers, Amer. Math. Monthly, 103 (1996) 729–741.
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comput. 16(1962) 363–367; MR 26 #6139.
Christoph Baxa, Über Gandhis Primzahlformel, Elem. Math., 47(1992) 8284; MR 93h: 1 1007.
E. Bedocchi, Nota ad una congettura sui numeri primi, Riv. Mat. Univ. Parma(4), 11 (1985) 229–236.
David Borwein, Jonathan M. Borwein, Peter B. Borwein and Roland Girgensohn, Giuga’s conjecture on primality, Amer. Math. Monthly, 103(1996) 40–50; MR 97b: 1 1004.
Jonathan M. Borwein and Erick Bryce Wong, A survey of results relating to Giuga’s conjecture on primality, Advances in Mathematical Sciences; CRM’s 25 years (Montreal, P.Q., 1994) 13–27, CRM Proc. Lecture Notes, 11, Amer. Math. Soc., Providence RI, 1997; MR 98i: 1 1005.
Nigel Boston and Marshall L. Greenwood, Quadratics representing primes, Amer. Math. Monthly, 102(1995) 595–599; MR 96g: 1 1154.
J. Braun, Das Fortschreitungsgesetz der Primzahlen durch eine transcendente Gleichung exakt dargestelt, Wiss. Beilage Jahresber. Fr. W. Gymn. Trier, 1899, 96 pp.
Rodrigo de Castro Korgi, Myths and realities about formulas for calculating primes (Spanish), Lect. Math., 14(1993) 77–101; MR 95c: 1 1009.
R. Creighton Buck, Prime representing functions, Amer. Math. Monthly, 53 (1946) 265.
John H. Conway, Problem 2.4, Math. Intelligencer, 3 (1980) 45.
L. E. Dickson, History of the Theory of Numbers, Carnegie Institute, Washington, 1919, 1920, 1923; reprinted Stechert, New York, 1934; Chelsea, New York, 1952, 1966, Vol. I, Chap. XV III.
Underwood Dudley, History of a formula for primes, Amer. Math. Monthly, 76(1969) 23 28; MR 38 #4270.
Underwood Dudley, Formulas for primes, Math. Mag., 56 (1983) 17–22.
D. D. Elliott, A prime generating function, Two-Year Coll. Math. J., 14 (1983) 57.
David Ellis, Some consequences of Wilson’s theorem, Univ. Nac. Tucumân Rev. Ser. A, 12(1959) 27–29; MR 21 #7179.
P. Erdös, A theorem of Sylvester and Schur, J. London Math. Soc., 9(1934) 282–288; Zbl. 10, 103.
P. Erdös, On consecutive integers, Nieuw Arch. Wisk. (3), 3(1955) 124–128; MR 17, 461f. [For the Fung and Williams reference, see A1.]
Reijo Ernvall, A formula for the least prime greater than a given integer, Elem. Math., 30(1975) 13–14; MR 54 #12616.
Robin Forman, Sequences with many primes, Amer. Math. Monthly 99(1992) 548–557; MR 93e: 1 1104.
J. M. Gandhi, Formulae for the n-th prime, Proc. Washington State Univ. Conf. Number Theory, Pullman, 1971, 96–106; MR 48 #218.
Betty Garrison, Polynomials with large numbers of prime values, Amer. Math. Monthly, 97(1990) 316–317; MR 91i: 1 1124.
James Gawlik, A function which distinguishes the prime numbers, Math. Today (Southend-on-Sea) 33 (1997) 21.
Giuseppe Giuga, Sopra alcune proprietà caratteristiche dei numeri primi, Period. Math. (4), 23(1943) 12–27; MR 8, 11.
Giuseppe Giuga, Su una presumibile proprietà caratteristica dei numeri primi, Ist. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat.(3), 14(83)(1950) 511–528; MR 13, 725.
P. Goetgheluck, On cubic polynomials giving many primes, Elem. Math., 44(1989) 70–73; MR 90j: 1 1014.
Solomon W. Golomb, A direct interpretation of Gandhi’s formula, Amer. Math. Monthly, 81(1974) 752–754; MR 50 #7003.
Solomon W. Golomb, Formulas for the next prime, Pacific J. Math. 63(1976) 401–404; MR 53 #13094.
R. L. Goodstein, Formulae for primes, Math. Gaz., 51 (1967) 35–36.
H. W. Gould, A new primality criterion of Mann and Shanks, Fibonacci Quart., 10(1972) 355–364, 372; MR 47 #119.
S. W. Graham, The greatest prime factor of the integers in an interval, J. London Math. Soc. (2), 24(1981) 427–440; MR 82k:10054.
Andrew Granville, Unexpected irregularities in the distribution of prime numbers, Proc. Internat. Congr. Math., Vol. 1, 2 (Zurich, 1994) 388–399, Birkhäuser Basel, 1995; MR 97d: 1 1139.
Andrew Granville and Sun Zhi-Wei, Values of Bernoulli polynomials, Pacific J. Math. 172(1996) 117–137; MR 98b:11018.
Silvio Guiasu, Is there any regularity in the distribution of prime numbers at the beginning of the sequence of positive integers? Math. Mag., 68 (1995) 110–121.
Richard K. Guy, Conway’s prime producing machine, Math. Mag., 56(1983) 26–33; MR 84j:10008.
G. H. Hardy, A formula for the prime factors of any number, Messenger of Math., 35 (1906) 145–446.
V. C. Harris, A test for primality, Nordisk Mat. Tidskr., 17(1969) 82; MR 40 #4197.
E. Härtter, Über die Verallgemeinerung eines Satzes von Sierpinski, Elem. Math. 16 (1961) 123–127; MR 24 #A1869.
Olga Higgins, Another long string of primes, J. Recreational Math. 14 (1981/82) 185.
C. Isenkrahe, Ueber eine Lösung der Aufgabe, jede Primzahl als Function der vorhergehenden Primzahlen durch einen geschlossenen Ausdruck darzustellen, Math. Ann., 53 (1900) 42–44.
Jia Chao-Hua, The greatest prime factor of the integers in a short interval, Acta Math. Sinica, 29(1986) 815–825; MR 88f:11083; II, 32(1989) 188–199; MR 90m:11135; III, ibid. (N.S.), 9(1993) 321–336; MR 95c: 1 1107.
James P. Jones, Formula for the n-th prime number, Canad. Math. Bull. 18(1975) 433–434; MR 57 #9641.
James P. Jones, Daihachiro Sato, Hideo Wada Si Douglas Wiens, Diophantine representation of the set of prime numbers, Amer. Math. Monthly 83(1976) 449464; MR 54 #2615.
James P. Jones and Yuri V. Matiyasevich, Proof of recursive unsolvability of Hilbert’s tenth problem, Amer. Math. Monthly, 98(1991) 689–709; MR 92i: 03050.
Matti Jutila, On numbers with a large prime factor, J. Indian Math. Soc. (N.S.), 37(1973) 43–53(1974); MR 50 #12936.
Steven Kahan, On the smallest prime greater than a given positive integer, Math. Mag., 47(1974) 91–93; MR 48 #10964.
Sushil Kumar Karmakar, An algorithm for generating prime numbers, Math. Ed. (Siwan), 28(1994) 175.
E. Karst, The congruence 2P-1 w 1 mod p2 and quadratic forms with a high density of primes, Elem. Math., 22 (1967) 85–88.
John Knopfmacher, Recursive formulae for prime numbers, Arch. Math. (Basel), 33 (1979/80) 144–149; MR 81j:10008.
Masaki Kobayashi, Prime producing quadratic polynomials and class-number one problem for real quadratic fields, Proc. Japan Acad. Ser. A Math. Sci., 66(1990) 119–121; MR 91i: 1 1140.
L. Kuipers, Prime-representing functions, Nederl. Akad. Wetensch. Proc., 53(1950) 309–310 = Indagationes Math., 12(1950) 57–58; MR 11, 644.
J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing ir(x): the Meissel Lehmer method, Math. Comput., 44 (1985) 537–560.
Klaus Langmann, Eine Formel für die Anzahl der Primzahlen, Arch. Math. (Basel), 25(1974) 40; MR 49 #4951.
A.-M. Legendre, Essai sur la Théorie des Nombres, Duprat, Paris, 1798, 1808.
D. H. Lehmer, On the function x2+x+A, Sphinx, 6(1936) 212–214; 7 (1937) 40.
Liu Hong-Kuan, The greatest prime factor of the integers in an interval, Acta Arith., 65(1993) 301–328; MR 95d:11117.
S. Louboutin, R. A. Mollin and H. C. Williams, Class numbers of real quadratic fields, continued fractions, reduced ideals, prime-producing polynomials and quadratic residue covers, Canad. J. Math., 44 (1992) 1–19.
H. B. Mann and Daniel Shanks, A necessary and sufficient condition for primality and its source, J. Combin. Theory Ser. A, 13(1972) 131–134; MR 46 #5225.
J.-P. Massias and G. Robin, Effective bounds for some functions involving prime numbers, Preprint, Laboratoire de Théorie des Nombres et Algorithmique, 123 rue A. Thomas, 87060 Limoges Cedex, France.
Yuri V. Matiyasevich, Primes are enumerated by a polynomial in 10 variables, Zap. Nauen. Sem. Leningrad. Otdel. Mat. Inst. Steklov,68(1977) 62–82, 144–145; MR 58 #21534; English translation: J. Soviet Math., 15 (1981) 33–44.
W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc., 53(1947) 604; MR 8, 567.
Richard A. Mollin, Prime valued polynomials and class numbers of quadratic fields, Internat. J. Math. Math. Sei., 13(1990) 1–11; MR 91c: 1 1060.
Richard A. Mollin, Prime-producing quadratics, Amer. Math. Monthly, 104(1997) 529–544; MR 98h: 1 1113.
Richard A. Mollin and Hugh Cowie Williams, Quadratic nonresidues and prime-producing polynomials, Canad. Math. Bull. 32(1989) 474–478; MR 91a:11009. [see also Number Theory de Gruyter, 1989, 654–663 and Nagoya Math. J. 112(1988) 143–151.]
Leo Moser, A prime-representing function, Math. Mag., 23 (1950) 163–164.
Joe L. Mott and Kermit Rose, Prime-producing cubic polynomials, Ideal theoretic methods in commutative algebra (Columbia MO 1999) 281–317, Lecture Notes in Pure and Appl. Math., 220 Dekker, New York, 2001; MR 2002c: 11140.
Gerry Myerson, Problems 96:16 and 96:17, Western Number Theory Problems, 96–12–16 and 19.
K S Namboodiripad, A note on formulae for the n-th prime, Monatsh. Math. 75(1971) 256–262; MR 46 #126.
T. B. M. Neill and M. Singer, The formula for the Nth prime, Math. Gaz., 49 (1965) 303.
Ivan Niven, Functions which represent prime numbers, Proc. Amer. Math. Soc., 2(1951) 753–755; MR 13, 321a.
O. Ore, On the selection of subsequences, Proc. Amer. Math. Soc., 3(1952) 706–712; MR 14, 256.
Joaquin Ortega Costa, The explicit formula for the prime number function 7r(x), Revista Mat. Hisp.-Amer.(4), 10(1950) 72–76; MR 12, 392b.
Laurentiu Panaitopol, On the inequality papb pab, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 41(89)(1998) 135–139; MR 2003c: 11115.
Makis Papadimitriou, A recursion formula for the sequence of odd primes, Amer. Math. Monthly 82(1975) 289; MR 52 #246.
Carlos Raitzin, The exact count of the prime numbers that do not exceed a given upper bound (Spanish), Rev. Ingr., 1(1979) 37–43; MR 82e:10074.
K. Ramachandra, A note on numbers with a large prime factor, J. London Math. Soc. (2), 1(1969) 303–306; MR 40 #118; II J, Indian Math. Soc. (N.S.) 34(1970) 39–48(1971); MR 45 #8616.
Stephen Regimbal, An explicit formula for the k-th prime number, Math. Mag., 48(1975) 230–232; MR 51 #12676.
Paulo Ribenboim, Selling primes, Math. Mag., 68 (1995) 175–182.
Paulo Ribenboim, Are there functions that generate prime numbers? College Math. J., 28(1997) 352–359; MR 98h: 1 1019.
B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Königl. Preuss. Akad. Wiss. Berlin 1859, 671–680; also in Gesammelte Mathematische Werke, Teubner, Leipzig, 1892, pp. 145–155.
J. B. Rosser Si L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., 6 (1962) 64–94.
Michael Rubinstein, A formula and a proof of the infinitude of the primes, Amer. Math. Monthly 100 (1993) 388–392.
W. Sierpinski, Elementary Number Theory, (ed. A. Schinzel) PWN, Warszawa, 1987, p. 218.
W. Sierpinski, Sur une formule donnant tous les nombres premiers, C.R. Acad. Sci. Paris, 235(1952) 1078–1079; MR 14, 355.
W. Sierpinski, Les binômes x2 + n et les nombres premiers, Bull. Soc. Royale Sciences Liège, 33 (1964) 259–260.
B. R. Srinivasan, Formulae for the n-th prime, J. Indian Math. Soc. (N.S.), 25(1961) 33–39; MR 26 #1289.
B. R. Srinivasan, An arithmetical function and an associated formula for the n-th prime. I, Norske Vid. Selsk. Forh. (Trondheim) 35(1962) 68–71; MR 27 #101.
Garry J. Tee, Simple analytic expressions for primes, and for prime pairs, New Zealand Math. Mag., 9(1972) 32–44; MR 45 #8601.
E. Teuffel, Eine Rekursionsformel für Primzahlen, Jber. Deutsch. Math. Verein., 57(1954) 34–36; MR 15, 685.
John Thompson, A method for finding primes, Amer. Math. Monthly, 60 (1953) 175; MR 14, 621.
P. G. Tsangaris and James P. Jones, An old theorem on the GCD and its application to primes, Fibonacci Quart., 30(1992) 194–198; MR 93e: 1 1004.
Charles Vanden Eynden, A proof of Gandhi’s formula for the n-th prime, Amer. Math. Monthly 79(1972) 625; MR 46 #3425.
E. Vantieghem, On a congruence only holding for primes, Indag. Math. (N.S.), 2(1991) 253–255; MR 92e: 1 1005.
C. P. Willans, On formulae for the Nth prime number, Math. Gaz., 48 (1964) 413–415.
Erick Wong, Computations on normal families of primes, M.Sc. thesis, Simon Fraser Univ., August 1997.
C. P. Wormell, Formulae for primes, Math. Gaz., 51 (1967) 36–38.
E. M. Wright, A prime representing function, Amer. Math. Monthly, 58(1951) 616–618; MR 13, 321b.
E. M. Wright, A class of representing functions, J. London Math. Soc., 29 (1954) 63–71; MR 15, 288d.
Don Zagier, Die ersten 50 Millionen Primzahlen, Beihefte zu Elemente der Mathematik 15, Birkhäuser, Basel 1977.
P. Erdös, Problems in number theory and combinatorics, Congr. Numer. XVIII, Proc. 6th Conf. Numer. Math., Manitoba, 1976, 35–58 (esp. p. 53); MR 80e: 10005.
Jörg Brüdern and Alberto Perelli, The addition of primes and power, Canad. J. Math., 48(1996) 512–526; MR 97h: 1 1118.
R. Brünner, A. Perelli and J. Pintz, The exceptional set for the sum of a prime and a square, Acta Math. Hunger., 53(1989) 347–365; MR 91b: 1 1104.
Chen Yong-Gao, On integers of the forms k — 2’ and k2 + 1, J. Number Theory 89(2001) 121–125; MR 2002b: 11020.
Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29(1975) 79–81; MR 51 #12758.
R. Crocker, On the sum of a prime and of two powers of two, Pacific J. Math., 36(1971) 103–107; MR 43 #3200.
P. Erd6s, On integers of the form 2’+ p and some related problems, Summa Brasil. Math., 2(1947–51) 113–123; MR 13, 437.
Patrick X. Gallagher, Primes and powers of 2, Inventiones Math., 29(1975) 125–142; MR 52 #315.
C. Hooley, Applications of Sieve Methods, Academic Press, 1974, Chap. VIII.
Koichi Kawada, On the representation of numbers as the sum of a prime and a k th power. Interdisciplinary studies on number theory (Japanese) Sûrikaisekikenkyûsho Kôkyüroku No. 837(1993) 92–100; MR 95k: 1 1129.
Li Hong-Ze, The exceptional set for the sum of a prime and a square, Acta Math. Hangar., 99(2003) 123–141; MR 2004c: 11184.
Donald E. G. Malm, A graph of primes, Math. Mag., 66(1993) 317–320; MR 94k: 1 1132.
Walter E. Mientka and Roger C. Weitzenkamp, On f-plentiful numbers, J. Combin. Theory, 7(1969) 374–377; MR 42 #3015.
W. Narkiewicz, On a conjecture of Erdös, Colloq. Math., 37(1977) 313–315; MR 58 #21971.
A. Perelli and A. Zaccagnini, On the sum of a prime and a k th power, Izv. Ross. Akad. Nauk Ser. Mat., 59(1995) 185–200; MR 96f: 1 1134.
A. de Polignac, Recherches nouvelles sur les nombres premiers, C. R. Acad. Sci. Paris, 29(1849) 397–401, 738–739.
Sun Zhi-Wei, On integers not of the form ±paf qb, Proc. Amer. Math. Soc., 128(2000) 997–1002; MR 2000i: 11157.
Sun Zhi-Wei and Le Mao-Hua, Integers not of the form c(2’ + 2) + p°, Acta Arith., 99(2001) 183–190; MR 2002e: 11043.
Sun Zhi-Wei and Yang Si-Man, A note on integers of the form 2’ + cp, Proc. Edinb. Math. Soc.(2) 45(2002) 155–160; MR 2002j: 11117.
Saburô Uchiyama and Masataka Yorinaga, Notes on a conjecture of P. Erdôs, I, II, Math. J. Okayama Univ. 19(1977) 129–140; 20(1978) 41–49; MR 56 #11929; 58 #570.
Mladen V. Vassilev-Missana, Note on “extraordinary primes”, Notes Number Theory Discrete Math., 1(1995) 111–113; MR 97g: 1 1004.
R. C. Vaughan, Some applications of Montgomery’s sieve, J. Number Theory 5(1973) 64–79; MR 49 #7222.
V. H. Vu, High order complementary bases of primes, Integers, 2(2002) Al2; MR 2003k: 11017.
Wang Tian-Ze, On the exceptional set for the equation n = p + k 2, Acta Math. Sinica (N.S.), 11(1995) 156–167; MR 97i: 1 1104.
A. Zaccagnini, Additive problems with prime numbers, Rend. Sem. Mat. Univ. Politec. Torino, 53(1995) 471–486; MR 98c: 1 1109
Peter Fletcher, William Lindgren and Carl Pomerance, Symmetric and asymmetric primes, J. Number Theory, 58(1996) 89–99; MR 97c: 1 1007.
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Guy, R.K. (2004). Prime Numbers. 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_2
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