The first occurrence of large gaps between successive primes
Author:
Richard P. Brent
Journal:
Math. Comp. 27 (1973), 959963
MSC:
Primary 10A20
MathSciNet review:
0330021
Fulltext PDF Free Access
Abstract 
References 
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Abstract: A table of the first occurrence of a string of composite numbers between two primes is given for and 301. All such strings between primes less than have been accounted for. The computation supports some conjectures on the distribution of these strings.
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R. P. Brent, Empirical Evidence for a Proposed Distribution of Small Prime Gaps, Technical Report CS 123, Computer Science Dept., Stanford Univ., Calif., 1969.
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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 and J.
E. Littlewood, Some problems of ‘Partitio numerorum’;
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44 (1923), no. 1, 1–70. MR
1555183, http://dx.doi.org/10.1007/BF02403921
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S. M. Johnson, "An elementary remark on maximal gaps between successive primes," Math. Comp., v. 19, 1965, pp. 675676.
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F. Jones, M.
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J. Lander and T.
R. Parkin, On first appearance of prime
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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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Karl
Prachar, Primzahlverteilung, SpringerVerlag,
BerlinGöttingenHeidelberg, 1957 (German). MR 0087685
(19,393b)
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Daniel
Shanks, On maximal gaps between successive
primes, Math. Comp. 18 (1964), 646–651. MR 0167472
(29 #4745), http://dx.doi.org/10.1090/S00255718196401674728
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S. Weintraub, "Distribution of primes between and ," 1971. Copy deposited in the UMT File and reviewed in Math. Comp., v. 26, 1972, p. 596.
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A. E. Western, "Note on the magnitude of the difference between successive primes," J. London Math. Soc., v. 9, 1934, pp. 276278.
 [1]
 K. I. Appel & J. B. Rosser, Table for Estimating Functions of Primes, IDACRD Technical Report Number 4, 1961, p. 102. (Reviewed in RMT 55, Math. Comp., v. 16, 1962, pp. 500501.)
 [2]
 R. P. Brent, Empirical Evidence for a Proposed Distribution of Small Prime Gaps, Technical Report CS 123, Computer Science Dept., Stanford Univ., Calif., 1969.
 [3]
 J. H. Cadwell, "Large intervals between consecutive primes," Math. Comp., v. 25, 1971, pp. 909913. MR 0299567 (45:8615)
 [4]
 H. Cramér, "On the order of magnitude of the difference between consecutive prime numbers," Acta Arith., v. 2, 1937, pp. 2346.
 [5]
 J. W. L. Glaisher, "On long successions of composite numbers," Messenger Math., v. 7, 1877, pp. 102, 171.
 [6]
 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.)
 [7]
 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
 [8]
 S. M. Johnson, "An elementary remark on maximal gaps between successive primes," Math. Comp., v. 19, 1965, pp. 675676.
 [9]
 M. F. Jones, M. Lal & W. J. Blundon, "Statistics on certain large primes," Math. Comp., v. 21, 1967, pp. 103107. MR 36 #3707. MR 0220655 (36:3707)
 [10]
 L. J. Lander & T. R. Parkin, "On first appearance of prime differences," Math. Comp., v. 21, 1967, pp. 483488. MR 37 #6237. MR 0230677 (37:6237)
 [11]
 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.
 [12]
 K. Pracher, Primzahlverteilung, SpringerVerlag, Berlin, 1957, pp. 154164. MR 19, 393. MR 0087685 (19:393b)
 [13]
 D. Shanks, "On maximal gaps between successive primes," Math. Comp., v. 18, 1964, pp. 646651. MR 29 #4745. MR 0167472 (29:4745)
 [14]
 S. Weintraub, "Distribution of primes between and ," 1971. Copy deposited in the UMT File and reviewed in Math. Comp., v. 26, 1972, p. 596.
 [15]
 A. E. Western, "Note on the magnitude of the difference between successive primes," J. London Math. Soc., v. 9, 1934, pp. 276278.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197303300210
PII:
S 00255718(1973)03300210
Keywords:
Prime,
distribution of primes,
prime gap,
maximal prime gap,
successive composites,
consecutive primes
Article copyright:
© Copyright 1973
American Mathematical Society
