Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



The number of primes is finite

Author: Miodrag Živković
Journal: Math. Comp. 68 (1999), 403-409
MSC (1991): Primary 11B83; Secondary 11K31
MathSciNet review: 1484905
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a positive integer $n$ let $A_{n+1}=\sum _{i=1}^n (-1)^{n-i} i!,$ $!n = \sum _{i=0}^{n-1} i!$ and let $p_1=3612703$. The number of primes of the form $A_n$ is finite, because if $n\geq p_1$, then $A_n$ is divisible by $p_1$. The heuristic argument is given by which there exists a prime $p$ such that $p\mid !n$ for all large $n$; a computer check however shows that this prime has to be greater than $2^{23}$. The conjecture that the numbers $!n$ are squarefree is not true because ${54503^2}\mid {!26541}$.

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

  • K. Akiyama, Y. Kida, F. O’Hara, APRT–CLE, Cohen-Lenstra version of Adleman-Pomerance-Rumely Test, UBASIC program, 1988–1992.
  • G. Gogić, Parallel algorithms in arithmetic, Master thesis, Belgrade University, 1991.
  • Richard K. Guy, Unsolved problems in number theory, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. MR 1299330
  • A. Ivić and Ž. Mijajlović, On Kurepa’s problems in number theory, Publ. Inst. Math. (Beograd) (N.S.) 57(71) (1995), 19–28. Đuro Kurepa memorial volume. MR 1387351
  • Y. Kida, ECMX, Prime Factorization by ECM, UBASIC program, 1987–1990.
  • Donald E. Knuth, The art of computer programming. Vol. 2: Seminumerical algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969. MR 0286318
  • Đuro Kurepa, On the left factorial function $!n$, Math. Balkanica 1 (1971), 147–153. MR 286736
  • B. Malešević, Personal communication.
  • Ž. Mijajlović, On some formulas involving $!n$ and the verification of the $!n$-hypothesis by use of computers, Publ. Inst. Math. (Beograd) (N.S.) 47(61) (1990), 24–32. MR 1103525
  • Hans Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 897531
  • UBASIC, version 8.74, 1994.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 11B83, 11K31

Retrieve articles in all journals with MSC (1991): 11B83, 11K31

Additional Information

Miodrag Živković
Affiliation: Matematički Fakultet, Beograd

Keywords: Prime numbers, left factorial, divisibility
Received by editor(s): July 19, 1996
Received by editor(s) in revised form: January 23, 1997
Article copyright: © Copyright 1999 American Mathematical Society