Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

The first occurrence of large gaps between successive primes


Author: Richard P. Brent
Journal: Math. Comp. 27 (1973), 959-963
MSC: Primary 10A20
DOI: https://doi.org/10.1090/S0025-5718-1973-0330021-0
MathSciNet review: 0330021
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A table of the first occurrence of a string of $ 2r - 1$ composite numbers between two primes is given for $ r = 158(1)267,269,270,273,275,276,281,282,291,294$ and 301. All such strings between primes less than $ 2.6 \times 10^{12}$ have been accounted for. The computation supports some conjectures on the distribution of these strings.


References [Enhancements On Off] (What's this?)

  • [1] K. I. Appel & J. B. Rosser, Table for Estimating Functions of Primes, IDA-CRD Technical Report Number 4, 1961, p. 102. (Reviewed in RMT 55, Math. Comp., v. 16, 1962, pp. 500-501.)
  • [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. 909-913. 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. 23-46.
  • [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 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.)
  • [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. 1-70. MR 1555183
  • [8] S. M. Johnson, "An elementary remark on maximal gaps between successive primes," Math. Comp., v. 19, 1965, pp. 675-676.
  • [9] M. F. Jones, M. Lal & W. J. Blundon, "Statistics on certain large primes," Math. Comp., v. 21, 1967, pp. 103-107. 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. 483-488. 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. 56-57.
  • [12] K. Pracher, Primzahlverteilung, Springer-Verlag, Berlin, 1957, pp. 154-164. MR 19, 393. MR 0087685 (19:393b)
  • [13] D. Shanks, "On maximal gaps between successive primes," Math. Comp., v. 18, 1964, pp. 646-651. MR 29 #4745. MR 0167472 (29:4745)
  • [14] S. Weintraub, "Distribution of primes between $ {10^{14}}$ and $ {10^{14}} + {10^8}$," 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. 276-278.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10A20

Retrieve articles in all journals with MSC: 10A20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1973-0330021-0
Keywords: Prime, distribution of primes, prime gap, maximal prime gap, successive composites, consecutive primes
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society