The distribution of small gaps between successive primes
Author:
Richard P. Brent
Journal:
Math. Comp. 28 (1974), 315324
MSC:
Primary 1004; Secondary 10A25, 10H15
MathSciNet review:
0330017
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Abstract 
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Abstract: For and large N, a wellknown conjecture of Hardy and Littlewood implies that the number of primes such that is the least prime greater than p is asymptotic to where the are certain constants. We describe a method for computing these constants. Related constants are given to 10D for , and some empirical evidence supporting the conjecture is mentioned.
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 P. T. Bateman & R. A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers," Math Comp., v. 16, 1962, pp. 363367. MR 26 #6139. MR 0148632 (26:6139)
 [2]
 R. P. Brent, "The first occurrence of large gaps between successive primes," Math. Comp., v. 27, 1973, pp. 959963. MR 0330021 (48:8360)
 [3]
 F. Gruenberger & G. Armerding, Statistics on the First Six Million Prime Numbers, Paper P2460, The RAND Corporation, Santa Monica, Calif., 1961, 145 pp. (Copy deposited in the UMT File and reviewed in Math. Comp., v. 19, 1965, pp. 503505.)
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 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., v. 44, 1923, pp. 170. MR 1555183
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 M. F. Jones, M. Lal & W. J. Blundon, "Statistics on certain large primes," Math. Comp., v. 21, 1967, pp. 103107. (Corrigenda, v. 22, 1968, pp. 474 & 911.) MR 36 #3707. MR 0220655 (36:3707)
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 D. H. Lehmer, "Tables concerning the distribution of primes up to 37 millions", 1957. Copy deposited in the UMT File and reviewed in MTAC, v. 13, 1959, pp. 5657.
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 C. L. Liu, Introduction to Combinatorial Mathematics, McGrawHill, New York, 1968, Chap. 4. MR 38 #3154. MR 0234840 (38:3154)
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 B. H. Mayoh, "The second Goldbach conjecture revisited," Nordisk Tidskr. Informationsbehandling (BIT), v. 8, 1968, pp. 128133. MR 39 #125. MR 0238761 (39:125)
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 [10]
 J. W. Wrench, "Evaluation of Artin's constant and the twinprime constant," Math. Comp., v. 15, 1961, pp. 396398. MR 23 #A1619. MR 0124305 (23:A1619)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819740330017X
PII:
S 00255718(1974)0330017X
Keywords:
Prime,
distribution of primes,
HardyLittlewood conjecture,
prime gap,
twin primes
Article copyright:
© Copyright 1974
American Mathematical Society
