Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-14T17:26:39.061Z Has data issue: false hasContentIssue false
  • English
  • Français

On the prime factorization of binomial coefficients

Published online by Cambridge University Press:  09 April 2009

E. F. Ecklund Jr ,
R. B. Eggleton ,
P. Erdös  and
J. L. Selfridge
E. F. Ecklund Jr
Affiliation:
Department of Computer Science Oregon State UniversityCorvallis, Organ 97330, USA
R. B. Eggleton
Affiliation:
Department of Mathematics The University of Newcastle New south Wales 2308, Australia
P. Erdös
Affiliation:
Hungarian Academy of Sciences Reáltanoda U. 13-15 Budapest V Hungary
J. L. Selfridge
Affiliation:
Department of Mathematical Sciences Northern Illnois UniversityDeKalb, Illnois 60115, USA
Rights & Permissions [Opens in a new window]

Abstract

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 positive integers n and k, with n≥2k, let , where each prime factor of u is less than k, and each prime factor of v is at least equal to k. It is shown that u<v holds with just 12 exceptions, which are determined. If , where each prime factor of U is at most equal to k, and each prime factor of V is greater than k, then U<V holds with at most finitely many exceptions, 19 of which are determined. It is conjectured that there are no others.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 3 , November 1978 , pp. 257 - 269
Copyright
Copyright © Australian Mathematical Society 1978

References

Appel, K. I. and Rosser, J. Barkley (1961), Table for estimating functions of primes, Communications Research Division Tech. Report 4, Inst. for Defence Analyses, Princeton, N.J.Google Scholar
Ecklund, E. F. Jr, and Eggleton, R. B. (1972), “Prime factors of consecutive integers”, Amer. Math. Monthly 79, 1082–1089.Google Scholar
Erdös, P. (1934), “A theorem of Sylvester and Schur”, J. London Math. Soc. 9, 282–288.CrossRefGoogle Scholar
Erdös, P. and Graham, R. L. (1976), “On the prime factors of ”, Fibonacci Quart. 14, 348–352.Google Scholar
Lehmer, D. H. (1964), “On a problem of Størmer”, Illinois J. Math. 8, 57–79.Google Scholar
Mahler, K. (1961), Lectures on Diophantine Approximations I, University of Notre Dame. See Theorem (5, II), p. 159. (This is the most accessible reference, though not the earliest.)Google Scholar
Rosser, J. Barkley and Schoenfeld, L. (1962), “Approximate formulas for some functions of prime numbers”, Illinois J. Math. 6, 64–94.CrossRefGoogle Scholar
Rosser, J. Barkley and Schoenfeld, L. (1975), “Sharper bounds for the Chebyshev functions φ(x) and ψ(x)”, Math. Comp. 29, 243–269.Google Scholar
Schoenfeld, L. (1976), “Sharper bounds for the Chebyshev functions φ(x) and ψ(x), II”, Math. Comp. 30, 337–360.Google Scholar
Schur, I. (1929), “Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen”, S.B. Deutsch. Akad. Wiss. Berlin Kl. Math. Phys. Tech. 23, 1–24.Google Scholar
Størmer, G. (1897), Quelques Théorèmes sur l' Equation de Pell x2-Dy2 = ±1 et leurs Applications, Videnskabs-Selskabets Skrifter, Christiania, 2, 48pp.Google Scholar
Sylvester, J. J. (1892), “On arithmetical series”, Messenger Math. 21, 1–19, 87–120. (Collected Mathematical Papers, vol. 4, 1912, 687–731.)Google Scholar