A NOTE ON p-ADIC VALUATIONS OF SCHENKER SUMS

作者：Miska Piotr^*

来源：Colloquium Mathematicum, 2015, 140(1): 5-13.

DOI：10.4064/cm140-1-2

• p-adic valuation
• prime
• Schenker sum

A prime number p is called a Schenker prime if there exists n is an element of N+ such that p inverted iota n and p inverted iota a(n), where a(n) = Sigma(n)(j = 0) (n!/j!)n(j) is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of a n when p is a Schenker prime. In particular, they conjectured that for each k is an element of N+ there exists a unique positive integer n(k) < 5(k) such that v(5) (a(m.5)k + n(k)) >= k for each nonnegative integer m. We prove that for every k is an element of N+ the inequality v(5) (a(n)) >= k has exactly one solution modulo 5(k). This confirms the above conjecture. Moreover, we show that if 37 inverted iota n then v(37) (a(n)) <= 1, which disproves the other conjecture of the above mentioned authors.

• 出版日期2015