The number of primes is finite

Author:
Miodrag Zivkovic

Journal:
Math. Comp. **68** (1999), 403-409

MSC (1991):
Primary 11B83; Secondary 11K31

DOI:
https://doi.org/10.1090/S0025-5718-99-00990-4

MathSciNet review:
1484905

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a positive integer let and let . The number of primes of the form is finite, because if , then is divisible by . The heuristic argument is given by which there exists a prime such that for all large ; a computer check however shows that this prime has to be greater than . The conjecture that the numbers are squarefree is not true because .

**1.**K. Akiyama, Y. Kida, F. O'Hara, APRT-CLE, Cohen-Lenstra version of Adleman-Pomerance-Rumely Test, UBASIC program, 1988-1992.**2.**G. Gogi\'{c},*Parallel algorithms in arithmetic*, Master thesis, Belgrade University, 1991.**3.**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****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****5.**Y. Kida, ECMX, Prime Factorization by ECM, UBASIC program, 1987-1990.**6.**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****7.**Đuro Kurepa,*On the left factorial function !𝑛*, Math. Balkanica**1**(1971), 147–153. MR**286736****8.**B. Male\v{s}evi\'{c}, Personal communication.**9.**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****10.**Hans Riesel,*Prime numbers and computer methods for factorization*, Progress in Mathematics, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 1985. MR**897531****11.**UBASIC, version 8.74, 1994.

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

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

Additional Information

**Miodrag Zivkovic**

Affiliation:
Matematički Fakultet, Beograd

Email:
ezivkovm@matf.bg.ac.yu

DOI:
https://doi.org/10.1090/S0025-5718-99-00990-4

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