# Paul Erdős and the Difference of Primes

• János Pintz
Chapter
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 25)

## Abstract

In the present work we discuss several problems concerning the difference of primes, primarily regarding the difference of consecutive primes. Most of them were either initiated by Paul Erdős (sometimes with coauthors), or were raised ahead of Erdős; nevertheless he was among those who reached very important results in them (like the problem of the large and small gaps between consecutive primes).

## Keywords

Prime Number London Math Prime Divisor Arithmetic Progression Acta Arith
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
R. J. Backlund, Über die Differenzen zwischen den Zahlen, die zu den ersten n Primzahlen teilerfremd sind. Commentationes in honorem E. L. Lindelöf. Annales Acad. Sci. Fenn. 32 (1929), Nr. 2, 1–9.Google Scholar
2. [2]
R. C. Baker, G. Harman, J. Pintz, The difference between consecutive primes, II. Proc. London Math. Soc.(3) 83 (2001), no. 3, 532–562.
3. [3]
E. Bombieri, On the large sieve. Mathematika 12 (1965), 201–225.
4. [4]
E. Bombieri, H. Davenport, Small differences between prime numbers. Proc. Roy. Soc. Ser. A 293 (1966), 1–18.
5. [5]
J. Bourgain, On triples in arithmetic progressions. Geom. Funct. Anal. 9 (1999), no. 5, 968–984.
6. [6]
J. Bourgain, Roth’s theorem on progressions revisited. J. Anal. Math. 104 (2008), 155–192.
7. [7]
A. Brauer, H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre. Sber. Berliner Math. Ges. 29 (1930), 116–125.Google Scholar
8. [8]
Chen Jing Run, On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao 17 (1966), 385–386 (Chinese).Google Scholar
9. [9]
Chen Jing Run, On the representation of a large even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16 (1973), 157–176.
10. [10]
J. G. van der Corput, Sur l’hypothése de Goldbach pour presque tous les nombres pairs. Acta Arith. 2 (1937), 266–290.Google Scholar
11. [11]
J. G. van der Corput, Über Summen von Primzahlen und Primzahlquadraten. Math. Ann. 116 (1939), 1–50.
12. [12]
H. Cramér, Prime numbers and probability. 8. Skand. Math. Kongr., Stockholm, 1935, 107–115.Google Scholar
13. [13]
H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2 (1936), 23–46.Google Scholar
14. [14]
N. G. Čudakov, On the density of the set of even numbers which are not representable as a sum of two primes. Izv. Akad. Nauk. SSSR 2 (1938), 25–40.Google Scholar
15. [15]
L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers. Messenger of Math. (2), 33 (1904), 155–161.Google Scholar
16. [16]
P. D. T. A. Elliott, H. Halberstam, A conjecture in prime number theory. Symposia Mathematica Vol. 4 (1970) (INDAM, Rome, 1968/69), 59–72, Academic Press, London.Google Scholar
17. [17]
P. Erdős, On the difference of consecutive primes. Quart. J. Math. Oxford ser. 6 (1935), 124–128.
18. [18]
P. Erdős, The difference of consecutive primes. Duke Math. J. 6 (1940), 438–441.
19. [19]
P. Erdős, On the difference of consecutive primes. Bull. Amer. Math. Soc. (1948), 885–889.Google Scholar
20. [20]
P. Erdős, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 374–384.
21. [21]
P. Erdős, Problems and results on the differences of consecutive primes. Publ. Math. Debrecen 1 (1949), 33–37.
22. [22]
P. Erdős, Some problems on the distribution of prime numbers. Teoria dei Numeri, Math. Congr. Varenna, 1954, 8 pp., 1955.Google Scholar
23. [23]
P. Erdős, Some problems on Consecutive Prime Numbers. Mathematika 19 (1972), 91–95.
24. [24]
P. Erdős, Résultats et problèmes en théorie des nombres. Séminaire Delonge-Pisot-Poitou (14e annèe: 1972/73), Théorie des nombres, Fasc. 2. Exp. No. 24, 7 pp. Sécretariat Mathématique, Paris, 1973.Google Scholar
25. [25]
P. Erdős, Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975), Congress Numer. XVI, pp. 25–44, Utilitas Math., Winnipeg, Man., 1976.Google Scholar
26. [26]
P. Erdős, Problems in number theory and combinatorics. Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976), Congress. Numer. XVIII, pp. 35–58, Utilitas Math., Winnipeg, Man., 1977.Google Scholar
27. [27]
P. Erdős, Some personal reminiscences of the mathematical work of Paul Turán. Acta Arith. 37 (1980), 3–8.Google Scholar
28. [28]
P. Erdős, Many old and on some new problems of mine in number theory, Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Univ Manitoba, Winnipeg, Man., 1980), Congress. Numer. 30 (1981), 3–27.Google Scholar
29. [29]
P. Erdős, Some problems on Number Theory, in: Analytic and elementary number theory (Marseille, 1983). Publ. Math. Orsay, 86-1, pp. 53–67, Univ. Paris XI, Orsay, 1986.Google Scholar
30. [30]
P. Erdős, L. Mirsky, The distribution of values of the divisor function d(n). Proc. London Math. Soc. (3) 2 (1952), 257–271.
31. [31]
P. Erdős, P. Turán, On some sequences of integers. J. London Math. Soc. 11 (1936), 261–264.
32. [32]
P. Erdős, P. Turán, On some new questions on the distribution of prime numbers. Bull. Amer. Math. Soc. 54 (1948), 371–378.
33. [33]
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), 307–14.
34. [34]
35. [35]
H. Furstenberg, Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Analyse Math. 31 (1977), 204–256.
36. [36]
D. A. Goldston, C. Yıldırım, Higher correlations of divisor sums related to primes. III. Small gaps between primes. Proc. London Math. Soc. (3) 95 (2007), no. 3, 653–686.
37. [37]
D. A. Goldston, J. Pintz, C. Yıldırım, Primes in Tuples III: On the difference p n+v−pn. Funct. Approx. Comment. Math. 35 (2006), 79–89.
38. [38]
D. A. Goldston, J. Pintz, C. Yıldırım, Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862.
39. [39]
D. A. Goldston, J. Pintz, C. Yıldırım, Primes in tuples. II. Acta Math. 204 (2010), no. 1, 1–47.
40. [40]
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yıldırım, Small gaps between products of two primes. Proc. London Math. Soc. (3) 98 (2009), no. 3, 741–774.
41. [41]
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yıldırım, Small gaps between almost primes, the parity problem, and some conjectures of Erdős on consecutive integers. Int. Math. Res. Not. IMRN 2011, no. 7, 1439–1450.Google Scholar
42. [42]
W. T. Gowers, A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8 (1998), no. 3, 529–551.
43. [43]
W. T. Gowers, A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11 (2001), no. 3, 465–588.
44. [44]
A. Granville, Unexpected irregularities in the distribution of prime numbers, in: Proceedings of the International Congress of Mathematicians (Zürich, 1994). Vol. 1, 2, 388–399, Birkhäuser, Basel, 1995.Google Scholar
45. [45]
A. Granville, Harald Cramér and the Distribution of Prime Numbers. Scand. Actuarial J. No. 1 (1995), 12–28.Google Scholar
46. [46]
G. Greaves, Sieves in Number Theory. Springer, 2001.Google Scholar
47. [47]
B. Grein, T. Tao, The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) 167 (2008), no. 2, 481–547.
48. [48]
G. H. Hardy, 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.
49. [49]
G. H. Hardy, J. E. Littlewood, Some problems of ‘Partitio Numerorum’, V: A further contribution to the study of Goldbach’s problem. Proc. London Math. Soc. (2) 22 (1924), 46–56.
50. [50]
D. R. Heath-Brown, Three primes and an almost prime in arithmetic progression. J. London Math. Soc. (2) 23 (1981), no. 3, 396–414.
51. [51]
D. R. Heath-Brown, The divisor function at consecutive integers. Mathematika 31 (1984), 141–149.
52. [52]
D. R. Heath-Brown, Integer sets containing no arithmetic progressions. J. London Math. Soc. (2) 35 (1987), no. 3, 385–394.
53. [53]
A. J. Hildebrand, H. Maier, Irregularities in the distribution of primes in short intervals. J. Reine Angew. Math. 397 (1989), 162–193.
54. [54]
G. Hoheisel, Primzahlprobleme in der Analysis. SBer. Preuss. Akad. Wiss., Berlin (1930), 580–588.Google Scholar
55. [55]
M. N. Huxley, On the differences of primes in arithmetical progressions. Acta Arith. 15 (1968/69), 367–392.
56. [56]
M. N. Huxley, Small differences between consecutive primes II. Mathematika 24 (1977), 142–152.
57. [57]
Chaohua Jia, Almost all short intervals containing prime numbers. Acta Arith. 76 (1996), 21–84.
58. [58]
J. Kaczorowski, A. Perelli, J. Pintz, A note on the exceptional set for Goldbach’s problem in short intervals. Monatsh. Math. 116, no. 3-4 (1993), 275–282. Corrigendum: ibid. 119 (1995), 215–216.
59. [59]
I. Kátai, A remark on a paper of Ju. V. Linnik. Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 17 (1967), 99–100.
60. [60]
L. Kronecker, Vorlesungen über Zahlentheorie, I. p. 68, Teubner, Leipzig, 1901.Google Scholar
61. [61]
E. Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion. Jahresber. Deutsche Math. Ver. 21 (1912), 208–228. [Proc. 5th Internat. Congress of Math., I, 93–108, Cambridge 1913; Collected Works, 5, 240–255, Thales Verlag.]
62. [62]
Yu. V. Linnik, Some conditional theorems concerning the binary Goldbach problem. Izv. Akad. Nauk. SSSR 16 (1952), 503–520 (Russian).
63. [63]
H. Maier, Chains of large gaps between consecutive primes. Adv. in Math. 39 (1981), no. 3, 257–269.
64. [64]
H. Maier, Small differences between prime numbers. Michigan Math. J. 35 (1988), 323–344.
65. [65]
H. Maier, C. Pomerance, Unusually large gaps between consecutive primes. Trans. Amer. Math. Soc. 322 (1990), 201–237.
66. [66]
E. Maillet, L’intermédiaire des math. 12 (1905), p. 108.Google Scholar
67. [67]
W. Narkiewicz, The Development of Prime Number Theory. From Euclid to Hardy and Littlewood. Springer, 2000.Google Scholar
68. [68]
J. Pintz, Very large gaps between consecutive primes. J. Number Th. 63 (1997), 286–301.
69. [69]
J. Pintz, Cramér vs. Cramér. On Cramér’s probabilistic model for primes. Funct. Approx. Comment. Math. 37 (2007), part 2, 361–376.
70. [70]
J. Pintz, Are there arbitrarily long arithmetic progressions in the sequence of twin primes? An irregular mind, Bolyai Soc. Math. Stud. 21, Springer, 2010, pp. 525–559.
71. [72]
J. Pintz, Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II. Proceedings of the Steklov Institute 276 (2012), 227–232.
72. [73]
A. de Polignac, Six propositions arithmologiques déduites de crible d’Ératosthene. Nouv. Ann. Math. 8 (1849), 423–429.Google Scholar
73. [74]
R. A. Rankin, The difference between consecutive prime numbers. J. London Math. Soc. 13 (1938), 242–244.
74. [75]
R. A. Rankin, The difference between consecutive prime numbers. II. Proc. Cambridge Philos. Soc. 36 (1940), 255–266.
75. [76]
R. A. Rankin, The difference between consecutive primes. III. J. London Math. Soc. 22 (1947), 226–230.
76. [77]
R. A. Rankin, The difference between consecutive prime numbers, V. Proc. Edinburgh Math. Soc. (2) 13 (1962/63), 331–332.
77. [78]
G. Ricci, Ricerche aritmetiche sui polinomi, II. (Intorno a una proposizione non vera di Legendre). Rend. Palermo 58 (1934), 190–208.
78. [79]
G. Ricci, La differenza di numeri primi consecutivi. Rendiconti Sem. Mat. Univ. e Politecnico Torino 11 (1952), 149–200. Corr. ibidem 12 (1953), p. 315.Google Scholar
79. [80]
G. Ricci, Sull’andamento della differenza di numeri primi consecutivi. Riv. Mat. Univ. Parma 5 (1954), 3–54.
80. [81]
K. F. Roth, Sur quelques ensembles d’entiers. C.R. Acad. Sci. Paris 234 (1952), 388–390.
81. [82]
K. F. Roth, On certain sets of integers. J. London Math. Soc. 28 (1953), 104–109.
82. [84]
T. Sanders, On Roth’s theorem on progressions. Ann. of Math. (2) 174 (2011), no. 1, 619–636.
83. [85]
A. Schönhage, Eine Bemerkung zur Konstruktion grosser Primzahllücken. Arch. Math. Basel 14 (1963), 29–30.
84. [86]
A. Selberg, An elementary proof of the prime-number theorem. Ann. of Math. (2) 50 (1949), 305–313.
85. [87]
J.-C. Schlage-Puchta, The equation ω(n) = ω(n + 1). Mathematika 50 (2003), no. 1-2, 99–101 (2005).
86. [88]
C. Spiro, Thesis. Urbana, 1981.Google Scholar
87. [89]
E. Szemerédi, On sets of integers containing no four elements in arithmetic progression. 1970 Number Theory (Colloq. János Bolyai Math. Soc. Debrecen, 1968), pp. 197–204, North-Holland, Amsterdam.Google Scholar
88. [90]
E. Szemerédi, On sets of integers containing no four elements in arithmetic progression. Acta Math. Acad. Sci. Hungar. 20 (1969), 89–104.
89. [91]
E. Szemerédi, On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 (1975), 199–245.
90. [92]
E. Szemerédi, Regular partitions of graphs. Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976) Colloq. Internat. CNRS 260, CNRS, Paris, 1978, pp. 399–401.Google Scholar
91. [93]
E. Szemerédi, Integer sets containing no arithmetic progressions. Acta Math. Hungar. 56 (1990), no. 1-2, 155–158.
92. [94]
A. I. Vinogradov, The density hypothesis for Dirichlet L-series. Izv. Akad. Nauk. SSSR 29 (1965), 903–934 (Russian). Corr.: ibidem, 30 (1966), 719–720.
93. [95]
I. M. Vinogradov, Representation of an odd number as a sum of three prime numbers. Doklady Akad. Nauk. SSSR 15 (1937), 291–294 (Russian).Google Scholar
94. [96]
I. M. Vinogradov, Special Variants of the Method of Trigonometric Sums. Nauka, Moskva, 1976 (Russian).Google Scholar
95. [97]
Wang Yuan, Xie Sheng-gang, Yu Kun-rui, Remarks on the difference of consecutive primes. Sci. Sinica 14 (1965), 786–788.
96. [98]
E. Westzynthius, Über die Verteilung der Zahlen, die zu der n ersten Primzahlen teilerfremd sind. Comm. Phys. Math. Helsingfors (5) 25 (1931), 1–37.Google Scholar