Hostname: page-component-54dcc4c588-tfzs5 Total loading time: 0 Render date: 2025-09-28T15:39:26.752Z Has data issue: false hasContentIssue false
  • English
  • Français

Consecutive integers with no large prime factors

Published online by Cambridge University Press:  09 April 2009

R. B. Eggleton  and
J. L. Selfridge
J. L. Selfridge
Affiliation:
Department of Mathematical SciencesNorthern Illinois UniversityDeKalb, Illinois 60115, U.S.A.
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.

For fixed integers k and m, with k ≥m ≥ 2, there are only finitely many runs of m consecutive integers with no prime factor exceeding k. We obatin lower bounds for the last such run. Let g(k, m) be its smallest member. For 2 ≤ m ≤ 5 it is shown that g(k, m) > kc logloglogk holds for all sufficiently large k, where c is a constant depending only on m. We also obtain a number of lower bounds with explict ranges of validity. A typical result of this type g(k, 3) > k3 holds just if k ≥ 41.

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 1 , August 1976 , pp. 1 - 11
Copyright
Copyright © Australian Mathematical Society 1976

References

Cunningham, A. J. C. and Woodall, H. J. (1925), Factorisation of , (Francis Hodgson, London, 1925).Google Scholar
Dickson, L. E. (1920), History of the theory of numbers, Vol. 2 (Carnegie Institution, 1920).Google Scholar
Ecklund, E. F. Jr, and Eggleton, R. B. (1972) ‘Prime factors of consecutive integers’, Amer. Math. Monthly 79, 1082–1089.CrossRefGoogle Scholar
Ecklund, E. F. Jr, Eggleton, R. B. and Selfridge, J. L. (1973), ‘Consecutive integers all of whose prime factors belong to a given set’, Proc. Third Manitoba Conf. on Numerical Math., Univ. of Manitoba, Winnipeg, 10 3 – 6, 1973, pp. 161–162.Google Scholar
Erdös, P. (1955), ‘On consecutive integers’, Nieuw Arch. Wisk. 3, 124–128.Google Scholar
Hardy, G. H. and Wright, E. M. (1960), An introduction to the theory of numbers (Clarendon, Oxford, 4th edn., 1960).Google Scholar
Lehmer, D. H. (1964), ‘On a problem of Stθrmer’, Illinois J. Math. 8, 57–79.Google Scholar
Lehmer, D. H. (1965), ‘The prime factors of consecutive integers’, Amer. Math. Monthly 72, no. 2, part II, 19–20.Google Scholar
Stθrmer, G. (1897), ‘Quelques théorèmes sur l'equation de Pell x2 – Dy2 = ± 1 et leurs applications,’ Videnskabs-Selskabets Skrifter, Christiania, 1897, no. 2, 48 pp.Google Scholar