On a problem of A. Rotkiewicz

On a problem of A. Rotkiewicz
HTML articles powered by AMS MathViewer

by Péter Kiss and Bui Minh Phong PDF

Math. Comp. 48 (1987), 751-755 Request permission

For any fixed positive integers a, $k \geqslant 2$ there are infinitely many composite integers n such that ${a^{n - k}} \equiv 1\;(\bmod n)$.

References

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
A. Mąkowski, Generalization of Morrow’s $D$ numbers, Simon Stevin 36 (1962/63), 71. MR 146136
D. C. Morrow, Mathematical Notes: Some Properties of D Numbers, Amer. Math. Monthly 58 (1951), no. 5, 329–330. MR 1527863, DOI 10.2307/2307722
Dragoslav S. Mitrinović and Milan S. Popadić, Inequalities in number theory, University of Niš, Department of Mathematics, Niš, 1978. MR 525579
A. Rotkiewicz, Pseudoprime numbers and their generalizations, University of Novi Sad, Student Association of the Faculty of Sciences, Novi Sad, 1972. MR 0330034
A. Rotkiewicz, On the congruence $2^{n-2}\equiv 1(\textrm {mod\,}n)$, Math. Comp. 43 (1984), no. 167, 271–272. MR 744937, DOI 10.1090/S0025-5718-1984-0744937-1
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265–284 (German). MR 1546236, DOI 10.1007/BF01692444