# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## The number of primes is finiteHTML articles powered by AMS MathViewer

by Miodrag Živković
Math. Comp. 68 (1999), 403-409 Request permission

## 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
• 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, DOI 10.1007/978-1-4899-3585-4
• 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, DOI 10.1007/978-1-4757-1089-2
• 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