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On the pseudoprimes of the form ax + b

  • A. Rotkiewicz (a1)
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A composite number n is called a pseudoprime if n|2n− 2.

Theorem 1. If a and b are natural numbers such that (a, b) = 1, then there exist infinitely many pseudoprimes of the form ax + b, where x is a natural number.

The proof of this theorem is given by the author in (5). This proof is based on the following two lemmas.

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References
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(1)Bang, A. S.Thaltheoretiske Undersogelser. Tidsskrift for Math. (5), 4 (1886), 7080, 130–137
(2)Birkhoff, G. D. and Vandiver, H. S.On the integral divisors of an − bn. Ann. of Math. (2), 5 (1904), 173180.
(3)Kanold, H. J.Sätze über Kristeilungspolynome und ihre Anwendungen and einige zahlentheoretische Probleme. J. für Math. 187 (1950), 169172.
(4)Prachar, K.Primzahlverteilung (Berlin–Gottingen–Heidelberg, 1957).
(5)Rotkiewicz, A.Sur les nombres pseudoprimes de la forme ax + b, C.R. Acad. Sci. Paris 257 (1963), 26012607.
(6)Schinzel, A.On primitive prime factors of an − bn, Proc. Cambridge Philos. Soc. 58 (1962), 555562.
(7)Zsigmondy, K.Zur Theorie der Potenzreste Monatsh. Math. 3 (1892), 265284.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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