On large Zsigmondy primes
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- by Walter Feit PDF
- Proc. Amer. Math. Soc. 102 (1988), 29-36 Request permission
Abstract:
If $a$ and $m$ are integers greater than 1, then a large Zsigmondy prime is a prime $l$ such that $l\left | {{a^m} - 1,l\nmid {a^i} - 1} \right .$ for $1 \leqslant i \leqslant m - 1$ and either ${l^2}\left | {{a^m} - 1} \right .$ or $l > m + 1$. The main result of this paper lists all the pairs $\left ( {a,m} \right )$ for which no large Zsigmondy prime exists.References
- Emil Artin, The orders of the linear groups, Comm. Pure Appl. Math. 8 (1955), 355–365. MR 70642, DOI 10.1002/cpa.3160080302
- Geo. D. Birkhoff and H. S. Vandiver, On the integral divisors of $a^n-b^n$, Ann. of Math. (2) 5 (1904), no. 4, 173–180. MR 1503541, DOI 10.2307/2007263 W. Feit, Extensions of cuspidal characters of ${\text {G}}{{\text {L}}_m}\left ( q \right )$ (to appear). W. Feit and G. Seitz (to appear).
- K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265–284 (German). MR 1546236, DOI 10.1007/BF01692444
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 29-36
- MSC: Primary 11A41
- DOI: https://doi.org/10.1090/S0002-9939-1988-0915710-1
- MathSciNet review: 915710