The prime number graph
Author:
Carl Pomerance
Journal:
Math. Comp. 33 (1979), 399408
MSC:
Primary 10A25; Secondary 52A10
MathSciNet review:
514836
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Abstract: Let denote the nth prime. The prime number graph is the set of lattice points , . We show that for every k there are k such points that are collinear. By considering the convex hull of the prime number graph, we show that there are infinitely many n such that for all positive . By a similar argument, we show that there are infinitely many n for which for all positive , thus verifying a conjecture of Selfridge. We make some new conjectures.
 [1]
P. ERDÖS, "On the difference of consecutive primes," Quart. J. Math. Oxford Ser., v. 6, 1935, pp. 124128.
 [2]
P.
Erdös, On the difference of consecutive
primes, Bull. Amer. Math. Soc. 54 (1948), 885–889. MR 0027009
(10,235b), http://dx.doi.org/10.1090/S000299041948090887
 [3]
P.
Erdős, Some applications of graph theory to number
theory, Proc. Second Chapel Hill Conf. on Combinatorial Mathematics
and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970)
Univ. North Carolina, Chapel Hill, N.C., 1970, pp. 136–145. MR 0266845
(42 #1748)
 [4]
Paul
Erdős, Problems and results on combinatorial number theory.
III, Number theory day (Proc. Conf., Rockefeller Univ., New York,
1976), Springer, Berlin, 1977, pp. 43–72. Lecture Notes in
Math., Vol. 626. MR 0472752
(57 #12442)
 [5]
P.
Erdős and K.
Prachar, Sätze und Probleme über
𝑝_{𝑘}/𝑘, Abh. Math. Sem. Univ. Hamburg
25 (1961/1962), 251–256 (German). MR 0140481
(25 #3901)
 [6]
P.
Erdös and A.
Rényi, Some problems and results on consecutive primes,
Simon Stevin 27 (1950), 115–125. MR 0034799
(11,644d)
 [7]
P.
Erdös and P.
Turán, On some new questions on the
distribution of prime numbers, Bull. Amer.
Math. Soc. 54
(1948), 371–378. MR 0024460
(9,498k), http://dx.doi.org/10.1090/S000299041948090103
 [8]
Douglas
Hensley and Ian
Richards, Primes in intervals, Acta Arith. 25
(1973/74), 375–391. MR 0396440
(53 #305)
 [9]
A.
E. Ingham, The distribution of prime numbers, Cambridge
Mathematical Library, Cambridge University Press, Cambridge, 1990. Reprint
of the 1932 original; With a foreword by R. C. Vaughan. MR 1074573
(91f:11064)
 [10]
L.
Thomas Ramsey, FourierStieltjes transforms of measures with a
certain continuity property, J. Functional Analysis
25 (1977), no. 3, 306–316. MR 0442601
(56 #982)
 [11]
Joseph
L. Gerver and L.
Thomas Ramsey, On certain sequences of lattice points, Pacific
J. Math. 83 (1979), no. 2, 357–363. MR 557936
(81c:10039)
 [12]
R.
A. Rankin, The difference between consecutive prime numbers.
V, Proc. Edinburgh Math. Soc. (2) 13 (1962/1963),
331–332. MR 0160767
(28 #3978)
 [13]
G.
J. Rieger, Über 𝑝_{𝑘}/𝑘 und verwandte
Folgen, J. Reine Angew. Math. 221 (1966), 14–19
(German). MR
0183698 (32 #1178)
 [14]
L. SCHOENFELD. (In preparation.)
 [15]
Sanford
L. Segal, On
𝜋(𝑥+𝑦)≤𝜋(𝑥)+𝜋(𝑦),
Trans. Amer. Math. Soc. 104 (1962), 523–527. MR 0139586
(25 #3018), http://dx.doi.org/10.1090/S00029947196201395864
 [16]
Sol
Weintraub, Seventeen primes in arithmetic
progression, Math. Comp.
31 (1977), no. 140, 1030. MR 0441849
(56 #240), http://dx.doi.org/10.1090/S00255718197704418494
 [1]
 P. ERDÖS, "On the difference of consecutive primes," Quart. J. Math. Oxford Ser., v. 6, 1935, pp. 124128.
 [2]
 P. ERDÖS, "On the difference of consecutive primes," Bull. Amer. Math. Soc., v. 54, 1948, pp. 885889. MR 0027009 (10:235b)
 [3]
 P. ERDÖS, Some Applications of Graph Theory to Number Theory, Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N. C., 1970), pp. 136145. Also see Some Recent Problems and Results in Graph Theory, Combinatorics and Number Theory, Proc. Seventh Southeastern Conf. on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1976), pp. 314. MR 0266845 (42:1748)
 [4]
 P. ERDÖS, Problems and Results on Combinatorial Number Theory. III, Number Theory Day (Proc. of the Conf. held at Rockefeller Univ., New York, 1976), Lecture Notes in Math., vol. 626, 1977, pp. 4372. MR 0472752 (57:12442)
 [5]
 P. ERDÖS & K. PRACHAR, "Satz und Probleme über ," Abh. Math. Sem. Univ. Hamburg, v. 25, 1961/62, pp. 251256. MR 0140481 (25:3901)
 [6]
 P. ERDÓS & A. RÉNYI, "Some problems and results on consecutive primes," Simon Stevin, v. 27, 1950, pp. 115125. MR 0034799 (11:644d)
 [7]
 P. ERDÖS & P. TURÁN, "On some new questions on the distribution of prime numbers," Bull. Amer. Math. Soc., v. 54, 1948, pp. 371378. MR 0024460 (9:498k)
 [8]
 D. HENSLEY & I. RICHARDS, "Primes in intervals," Acta Arith., v. 25, 1974, pp. 375391. MR 0396440 (53:305)
 [9]
 A. E. INGHAM, The Distribution of Prime Numbers, Cambridge Univ. Press, London, 1932. MR 1074573 (91f:11064)
 [10]
 L. T. RAMSEY, "FourierStieltjes transforms of measures with a certain continuity property," J. Functional Analysis, v. 25, 1977, pp. 306313. MR 0442601 (56:982)
 [11]
 L. T. RAMSEY & J. GERVER, "On certain sequences of lattice points," Pacific J. Math. (To appear.) MR 557936 (81c:10039)
 [12]
 R. A. RANKIN, "The difference between consecutive primes. V," Proc. Edinburgh Math. Soc. (2), v. 13, 1962/63, pp. 331332. MR 0160767 (28:3978)
 [13]
 G. J. RIEGER, "Über und verwandte Folgen," J. Reine Angew. Math., v. 221, 1966, pp. 1419. MR 0183698 (32:1178)
 [14]
 L. SCHOENFELD. (In preparation.)
 [15]
 S. L. SEGAL, "On ," Trans. Amer. Math. Soc., v. 104, 1962, pp. 523527. MR 0139586 (25:3018)
 [16]
 S. WEINTRAUB, "Seventeen primes in arithmetic progression," Math. Comp., v. 31, 1977, p. 1030. MR 0441849 (56:240)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905148367
PII:
S 00255718(1979)05148367
Article copyright:
© Copyright 1979
American Mathematical Society
