On the primality of 𝑛!±1 and 2×3×5×\dotsm×𝑝±1

Caldwell, Chris; Gallot, Yves

doi:10.1090/S0025-5718-01-01315-1

On the primality of $n! \pm 1$ and $2 \times 3 \times 5 \times \dotsm \times p \pm 1$
HTML articles powered by AMS MathViewer

by Chris K. Caldwell and Yves Gallot;

Math. Comp. 71 (2002), 441-448

DOI: https://doi.org/10.1090/S0025-5718-01-01315-1

Published electronically: May 11, 2001

PDF | Request permission

Abstract:

For each prime $p$, let $p\#$ be the product of the primes less than or equal to $p$. We have greatly extended the range for which the primality of $n! \pm 1$ and $p\# \pm 1$ are known and have found two new primes of the first form ($6380!+1, 6917!-1$) and one of the second ($42209\#+1$). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.

References

Eric Bach and Jeffrey Shallit, Algorithmic number theory. Vol. 1, Foundations of Computing Series, MIT Press, Cambridge, MA, 1996. Efficient algorithms. MR 1406794
D. Bailey, FFTs in external or hierarchical memory, Journal of Supercomputing 4:1 (1990), 23–35.
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363–367. MR 148632, DOI 10.1090/S0025-5718-1962-0148632-7
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, 441–446. With microfiche supplement. MR 1433262, DOI 10.1090/S0025-5718-98-00891-6
Alan Borning, Some results for $k\,!\pm 1$ and $2\cdot 3\cdot 5\cdots p\pm 1$, Math. Comp. 26 (1972), 567–570. MR 308018, DOI 10.1090/S0025-5718-1972-0308018-5
John Brillhart, D. H. Lehmer, and J. L. Selfridge, New primality criteria and factorizations of $2^{m}\pm 1$, Math. Comp. 29 (1975), 620–647. MR 384673, DOI 10.1090/S0025-5718-1975-0384673-1
Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
Joseph H. Silverman and John Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. MR 1171452, DOI 10.1007/978-1-4757-4252-7
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
Richard E. Crandall, Topics in advanced scientific computation, Springer-Verlag, New York; TELOS. The Electronic Library of Science, Santa Clara, CA, 1996. MR 1392472, DOI 10.1007/978-1-4612-2334-4
Richard Crandall, Karl Dilcher, and Carl Pomerance, A search for Wieferich and Wilson primes, Math. Comp. 66 (1997), no. 217, 433–449. MR 1372002, DOI 10.1090/S0025-5718-97-00791-6
H. Dubner, The development of a powerful low-cost computer for number theory applications, J. Recreational Math. 18 (1985-86), 81–86.
—, Factorial and primorial primes, J. Recreational Math. 19:3 (1987), 197–203.
—, A new primorial prime, J. Recreational Math. 21:4 (1989), 276.
Harvey Dubner, Large Sophie Germain primes, Math. Comp. 65 (1996), no. 213, 393–396. MR 1320893, DOI 10.1090/S0025-5718-96-00670-9
H. Dubner and Y. Gallot, Distribution of generalized Fermat prime numbers, Preprint, 1999.
Harvey Dubner and Wilfrid Keller, New Fibonacci and Lucas primes, Math. Comp. 68 (1999), no. 225, 417–427, S1–S12. MR 1484896, DOI 10.1090/S0025-5718-99-00981-3
Pierre Dusart, The $k$th prime is greater than $k(\ln k+\ln \ln k-1)$ for $k\geq 2$, Math. Comp. 68 (1999), no. 225, 411–415. MR 1620223, DOI 10.1090/S0025-5718-99-01037-6
Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
Y. Gallot, Proth.exe: a windows program for finding very large primes, 1999, http://www.utm.edu/research/primes/programs/gallot/.
G. H. Hardy, Collected papers of G. H. Hardy (Including Joint papers with J. E. Littlewood and others). Vol. I, Clarendon Press, Oxford, 1966. Edited by a committee appointed by the London Mathematical Society. MR 201267
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
Wilfrid Keller, New Cullen primes, Math. Comp. 64 (1995), no. 212, 1733–1741, S39–S46. With a biographical sketch of James Cullen by T. G. Holt and a supplement by Keller and Wolfgang Niebuhr. MR 1308456, DOI 10.1090/S0025-5718-1995-1308456-3
Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. MR 378456
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
P Mihailescu and C. Nash, Binary tree evaluation method for Lucas-Lehmer primality tests, preprint, 1999.
C. Nash, 42209#+1 is prime, personal communication to the authors, May 1999.
Manfred R. Schroeder, Where is the next Mersenne prime hiding?, Math. Intelligencer 5 (1983), no. 3, 31–33. MR 737688, DOI 10.1007/BF03026569
F. Proth, Théorèmes sur Les Nombres Premiers, C. R. Acad. Sci. Paris 85 (1877), 329–331.
Manfred R. Schroeder, Where is the next Mersenne prime hiding?, Math. Intelligencer 5 (1983), no. 3, 31–33. MR 737688, DOI 10.1007/BF03026569
Daniel Shanks, Solved and unsolved problems in number theory, 2nd ed., Chelsea Publishing Co., New York, 1978. MR 516658
Wacław Sierpiński, Elementary theory of numbers, Monografie Matematyczne [Mathematical Monographs], Tom 42, Państwowe Wydawnictwo Naukowe, Warsaw, 1964. Translated from Polish by A. Hulanicki. MR 175840
Mark Templer, On the primality of $k!+1$ and $2\ast 3$ $\ast 5\ast \cdots \ast \,p+1$, Math. Comp. 34 (1980), no. 149, 303–304. MR 551306, DOI 10.1090/S0025-5718-1980-0551306-2
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Numerical recipes in FORTRAN, 2nd ed., Cambridge University Press, Cambridge, 1992. The art of scientific computing; With a separately available computer disk. MR 1196230
Samuel S. Wagstaff Jr., Divisors of Mersenne numbers, Math. Comp. 40 (1983), no. 161, 385–397. MR 679454, DOI 10.1090/S0025-5718-1983-0679454-X