On multiple prime divisors of cyclotomic polynomials

Author:
Wayne L. McDaniel

Journal:
Math. Comp. **28** (1974), 847-850

MSC:
Primary 10A40; Secondary 10-04

DOI:
https://doi.org/10.1090/S0025-5718-1974-0387177-4

MathSciNet review:
0387177

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *q* be a prime $< 150$ and ${F_n}$ be the cyclotomic polynomial of order *n*. All triples (*p, n, q*) with *p* an odd prime $< {10^6}$ when $q < 100$ and $p < {10^4}$ when $100 < q < 150$ are given for which ${F_n}(q)$ is divisible by ${p^t}(t > 1)$.

- J. Brillhart, J. Tonascia, and P. Weinberger,
*On the Fermat quotient*, Computers in number theory (Proc. Sci. Res. Council Atlas Sympos. No. 2, Oxford, 1969) Academic Press, London, 1971, pp. 213–222. MR**0314736** - Trygve Nagell,
*Introduction to Number Theory*, John Wiley & Sons, Inc., New York; Almqvist & Wiksell, Stockholm, 1951. MR**0043111** - Hans Rademacher,
*Lectures on elementary number theory*, A Blaisdell Book in the Pure and Applied Sciences, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR**0170844** - Hans Riesel,
*Note on the congruence $a^{p-1}=1$ $({\rm mod}$ $p^{2})$*, Math. Comp.**18**(1964), 149–150. MR**157928**, DOI https://doi.org/10.1090/S0025-5718-1964-0157928-6
J. J. Sylvester, "On the divisors of the sum of a geometrical series whose first term is unity and common ratio any positive or negative number,"

*Nature*, v. 37, 1888, pp. 417-418;

*Collected Mathematical Papers*, v. 4, 1912, pp. 625-629.

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Keywords:
Cyclotomic polynomial,
sum of divisors

Article copyright:
© Copyright 1974
American Mathematical Society