Wilson, Brad Asymptotic behavior of Pascal’s triangle modulo a prime. (English) Zbl 0903.11006 Acta Arith. 83, No. 2, 105-116 (1998). Denote by \(F_p(n)\) the number of entries in the first \(n\) rows of Pascal’s triangle not divisible by a given prime number \(p\). For \(\vartheta_p= \ln\left( {p(p+1) \over 2} \right)/ \ln p\) let \(\alpha_p\) and \(\beta_p\) be defined as the upper resp. lower limit of the sequence \({F_p(n) \over n^{\theta_p}}\). H. Harborth [Proc. Am. Math. Soc. 62, 19-22 (1977; Zbl 0342.10040)] showed \(\alpha_2 =1\) and he determined \(\beta_2\) to six decimal places. A. H. Stein [in Number Theory, Lect. Notes Math. 1383, 170-177 (1989; Zbl 0692.10014)] proved \(\alpha_p=1\) for all primes \(p\). The author calculates \(\beta_3\), \(\beta_5, \dots, \beta_{19}\) to six decimals and he shows \(\lim_{p \to\infty} \beta_p= {1\over 2}\). Reviewer: Thomas Maxsein (Clausthal) Cited in 1 ReviewCited in 1 Document MSC: 11B65 Binomial coefficients; factorials; \(q\)-identities 05A10 Factorials, binomial coefficients, combinatorial functions 11B50 Sequences (mod \(m\)) Keywords:binomial coefficients; Kummer’s theorem; asymptotic behaviour; Pascal’s triangle Citations:Zbl 0342.10040; Zbl 0692.10014 PDFBibTeX XMLCite \textit{B. Wilson}, Acta Arith. 83, No. 2, 105--116 (1998; Zbl 0903.11006) Full Text: DOI EuDML