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

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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.**R. Guy,*Unsolved problems in number theory*, Second Edition, Springer-Verlag, 1994.MR**96e:11002****4.**A. Ivi\'{c}, \v{Z}. Mijajlovi\'{c},*On Kurepa's problems in number theory*, Publ. Inst. Math. (Beograd) (N. S.),**57(71)**, 1995, 19-28 MR**97a:11007****5.**Y. Kida, ECMX, Prime Factorization by ECM, UBASIC program, 1987-1990.**6.**D. E. Knuth,*The Art of Computer Programming*, Vol. 2 Addison-Wesley, Reading 1969.MR**44:3531****7.**Dj. Kurepa,*On the left factorial function*, Math. Balkanica,**1**, 1971, 147-153.MR**44:3945****8.**B. Male\v{s}evi\'{c}, Personal communication.**9.**\v{Z}. Mijajlovi\'{c},*On some formulas involving and the verification of the hypothesis by use of computers*, Publ. Inst. Math. (Beograd),**4**7(61), 1990, 24-32. MR**92d:11134****10.**H. Riesel,*Prime numbers and computer methods for factorization*, Birkhauser, Boston, 1985.MR**88k:11002****11.**UBASIC, version 8.74, 1994.

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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