The distribution of small gaps between successive primes

Author:
Richard P. Brent

Journal:
Math. Comp. **28** (1974), 315-324

MSC:
Primary 10-04; Secondary 10A25, 10H15

DOI:
https://doi.org/10.1090/S0025-5718-1974-0330017-X

MathSciNet review:
0330017

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Abstract | References | Similar Articles | Additional Information

Abstract: For and large *N*, a well-known conjecture of Hardy and Littlewood implies that the number of primes such that is the least prime greater than *p* is asymptotic to

**[1]**P. T. Bateman & R. A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers,"*Math Comp.*, v. 16, 1962, pp. 363-367. 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. 959-963. MR**0330021 (48:8360)****[3]**F. Gruenberger & G. Armerding,*Statistics on the First Six Million Prime Numbers*, Paper P-2460, The RAND Corporation, Santa Monica, Calif., 1961, 145 pp. (Copy deposited in the UMT File and reviewed in*Math. Comp.*, v. 19, 1965, pp. 503-505.)**[4]**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. 1-70. MR**1555183****[5]**M. F. Jones, M. Lal & W. J. Blundon, "Statistics on certain large primes,"*Math. Comp.*, v. 21, 1967, pp. 103-107. (Corrigenda, v. 22, 1968, pp. 474 & 911.) MR**36**#3707. MR**0220655 (36:3707)****[6]**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. 56-57.**[7]**C. L. Liu,*Introduction to Combinatorial Mathematics*, McGraw-Hill, New York, 1968, Chap. 4. MR**38**#3154. MR**0234840 (38:3154)****[8]**B. H. Mayoh, "The second Goldbach conjecture revisited,"*Nordisk Tidskr. Informationsbehandling*(*BIT*), v. 8, 1968, pp. 128-133. MR**39**#125. MR**0238761 (39:125)****[9]**H. Riesel, "Primes forming arithmetic series and clusters of large primes,"*Nordisk Tidskr. Informationsbehandling*(*BIT*), v. 10, 1970, pp. 333-342. MR**44**#965. MR**0282931 (44:165)****[10]**J. W. Wrench, "Evaluation of Artin's constant and the twin-prime constant,"*Math. Comp.*, v. 15, 1961, pp. 396-398. MR**23**#A1619. MR**0124305 (23:A1619)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0330017-X

Keywords:
Prime,
distribution of primes,
Hardy-Littlewood conjecture,
prime gap,
twin primes

Article copyright:
© Copyright 1974
American Mathematical Society