Hostname: page-component-68945f75b7-tmfhh Total loading time: 0 Render date: 2024-08-05T21:02:06.488Z Has data issue: false hasContentIssue false
  • English
  • Français

On the prime factors of the number 2p-1 - 1

Published online by Cambridge University Press:  18 May 2009

A. Rotkiewicz
A. Rotkiewicz
Affiliation:
Department of Pure MathematicsCambridge University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

From the proof of Theorem 2 of [5] it follows that for every positive integer k there exist infinitely many primes p in the arithmetical progression ax + b (x = 0, 1, 2,…), where a and b are relatively prime positive integers, such that the number 2p−1 − 1 has at least k composite factors of the form (p − 1)x + 1. The following question arises:

For any given natural number k, do there exist infinitely many primes p such that the number 2p−1 − 1 has k prime factors of the form(p − 1)x + 1 and p ≡ b (mod a), where a and b are coprime positive integers?

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 9 , Issue 2 , July 1968 , pp. 83 - 86
Copyright
Copyright © Glasgow Mathematical Journal Trust 1968

References

REFERENCES

1.Capelli, A., Sulla reduttibilita della funzione x n-A in un campo qualunque di razionalità, Math. Ann. 54 (1901), 602–603.CrossRefGoogle Scholar
2.Dedekind, R., Ueber den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Congruenzen (Göttingen, 1878).Google Scholar
3.Hasse, H., Bericht über neuere Untersuchungen and Probleme aus der Theorie der algebraischen Zahlen, Teil II, Reziprozitätsgesetze (Berlin, 1930).Google Scholar
4.Nagell, T., Contributions à la théorie des corps et des polynômes cyclotomiques, Ark. Mat. 5 (1965), 153–192.Google Scholar
5.Rotkiewicz, A., Sur les nombres naturels n et k tels que les nombres n et nk sont a la fois pseudoprimes, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fiz. Mat. Nat. (8) 36 (1964), 816–818.Google Scholar
6.Rotkiewicz, A., Sur les nombres premiers p et q tels que pq | 2pq − 2, Rend. Cire. Mat. Palermo 11 (1962), 280–282.CrossRefGoogle Scholar
7.Szymiczek, K., On prime numbers p, q and r such that pq, pr and qr are pseudoprimes, Colloq. Math. 13 (1965), 259–263.CrossRefGoogle Scholar
8.Schinzel, A.. On primitive factors of a n − b n, Proc. Cambridge Philos. Soc. (4), 58 (1962), 555–562.CrossRefGoogle Scholar
9.Zsigmondy, K., Zur Theorie der Potenzreste, Monatsh. Math. 3 (1892), 265–284.CrossRefGoogle Scholar