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On Some Divisibility Properties of (2nn)

Published online by Cambridge University Press:  20 November 2018

P. Erdös
P. Erdös*
Affiliation:
McGill University
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L. Moser [3] recently gave a very simple proof that

1.

has no solutions. In the present note we shall first of all prove that for , which by the fact that there is a prime p satisfying n < p ≤ 2n immediately implies that

2.

has no solutions. It is easy to see on the other hand that

3.

has infinitely many non-trivial solutions.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 7 , Issue 4 , December 1964 , pp. 513 - 518
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. See Erdős, P., Quelques Problèmes de la Thèorie des Nombres Problème 57, Monographie de L'Enseignement Math. No. 6.Google Scholar
2. Miss Faulkner' s proof is not yet published.Google Scholar
3. For the strongest result in this direction see Ingham, A. E., On the difference between consecutive primes, Quarterly Journal of Math. 8(1937), 255–266.CrossRefGoogle Scholar
4. Moser, L., Insolvability of Can. Math. Bull. 6 (1963) 167–169.CrossRefGoogle Scholar