Explicit estimates of the number of distinct prime divisors of binomial coefficients

Djamel Berkane; Boualem Sadaoui; Pierre Dusart

doi:10.3842/umzh.v76i8.7551

Djamel Berkane LAMDA-RO Laboratory, University of Blida1, Algeria https://orcid.org/0000-0001-9043-8468
Boualem Sadaoui LESI Laboratory, University of Khemis Miliana, Algeria https://orcid.org/0000-0002-4127-701X
Pierre Dusart XLIM UMR 7252, Faculté des sciences et techniques Université de Limoges, France https://orcid.org/0000-0003-2184-9022

DOI: https://doi.org/10.3842/umzh.v76i8.7551

Keywords: Binomial coefficient, Primorial number,, Explicit estimate, Prime divisor

Abstract

UDC 511

We propose explicit estimates of the number of distinct prime divisors of a binomial coefficient through the explicit generalizations of some principal existing results. We also prove interesting number-theoretical propositions for the terms of a sequence of primes.

References

H. Alzer, J. Sándor, On a binomial coefficient and a product of prime numbers, Appl. Anal. and Discrete Math., 5, 87–92 (2011). DOI: https://doi.org/10.2298/AADM110206008A

S. Casacuberta, On the divisibility of binomial coefficients, Ars Math. Contemp., 19, 297–309 (2020). DOI: https://doi.org/10.26493/1855-3974.2103.e84

E. F. Ecklund, P. Erdős, J. L. Selfridge, A new function associated with the prime factors of $begin{pmatrix}{n}{k}end{pmatrix}$, Math. Comp., 28, 647–649 (1974). DOI: https://doi.org/10.1090/S0025-5718-1974-0337732-2

P. Erdős, Über die Anzahl der Primfaktoren von $begin{pmatrix}{n}{k}end{pmatrix}$, Arch. Math., 24, 53–56 (1973). DOI: https://doi.org/10.1007/BF01228172

P. Erdős, R. L. Graham, I. Z. Ruzsa, E. G. Straus, On the prime factors of $begin{pmatrix}{2n}{n}end{pmatrix}$, Math. Comp., 29, 83–92 (1975). DOI: https://doi.org/10.1090/S0025-5718-1975-0369288-3

P. Erdős, H. Gupta, S. P. Khare, On the number of distinct prime divisors of $begin{pmatrix}{n}{k}end{pmatrix}$, Util. Math., 10, 51–60 (1976).

P. Erdős, Some unconventional problems in number theory, Math. Mag., 52, 67–70 (1979). DOI: https://doi.org/10.1080/0025570X.1979.11976756

P. Goetgheluck, Computing binomial coefficients, Amer. Math. Monthly, 94, 360–365 (1987). DOI: https://doi.org/10.1080/00029890.1987.12000648

P. Goetgheluck, On prime divisors of binomial coefficients, Math. Comp., 51, 325–329 (1988). DOI: https://doi.org/10.1090/S0025-5718-1988-0942159-6

H. Gupta, S. P. Khare, On $begin{pmatrix}{k^2}{k}end{pmatrix}$ and the product of the first $k$-primes, Publ. Fac. 'Electrotech. Univ. Belgrade, S'er. Math. Phys., 577-598, 25–29 (1977).

K. Marko, Divisibility of binomial coefficients near a half-line and in convex set, Acta Math. Univ. Comenian., 50/51, 267–275 (1987).

P. A. B. Pleasants, The number of prime factors of binomial coefficients, J. Number Theory, 15, 203–225 (1982). DOI: https://doi.org/10.1016/0022-314X(82)90026-9

G. Robin, Estimation de la fonction de Tchebychef $theta$ sur le $k^{text{ième}}$ nombre premier et grande valeurs de la fonction $omega(n)$ nombre de diviseurs premiers de $n$, Acta Arith., 42, 367–389 (1983). DOI: https://doi.org/10.4064/aa-42-4-367-389

A. Sárközy, On divisors of binomial coefficients. I, J. Number Theory, 20, 70–80 (1985). DOI: https://doi.org/10.1016/0022-314X(85)90017-4

H. Scheid, Die Anzahl der Primfaktoren in $begin{pmatrix}{n}{k}end{pmatrix}$, Arch. Math., 20, 581–582 (1969). DOI: https://doi.org/10.1007/BF01899057

N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences; https://oeis.org.

P. Stănică, Good lower and upper bounds on binomial coefficients, J. Inequal. Pure and Appl. Math., 2, Article 30 (2001).