Two new kinds of numbers and related divisibility results

Zhi-Wei Sun

paper

Two new kinds of numbers and related divisibility results

arXiv:1408.5381

Abstract

We mainly introduce two new kinds of numbers given by $R_{n} = k = 0 \sum n (k n) (k n + k) \frac{1}{2 k - 1} (n = 0, 1, 2, ...)$ and $S_{n} = k = 0 \sum n (k n)^{2} (k 2 k) (2 k + 1) (n = 0, 1, 2, ...) .$ We find that such numbers have many interesting arithmetic properties. For example, if $p \equiv 1 (mod 4)$ is a prime with $p = x^{2} + y^{2}$ (where $x \equiv 1 (mod 4)$ and $y \equiv 0 (mod 2)$ ), then $R_{(p - 1) /2} \equiv p - (- 1)^{(p - 1) /4} 2 x (mod p^{2}) .$ Also, $\frac{1}{n ^{2}} k = 0 \sum n - 1 S_{k} \in Z an d \frac{1}{n} k = 0 \sum n - 1 S_{k} (x) \in Z [x] for all n = 1, 2, 3, ...,$ where $S_{k} (x) = \sum_{j = 0}^{k} (j k)^{2} (j 2 j) (2 j + 1) x^{j}$ . For any positive integers $a$ and $n$ , we show that, somewhat surprisingly, $\frac{1}{n ^{2}} k = 0 \sum n - 1 (2 k + 1) (k n - 1)^{a} (k - n - 1)^{a} \in Z an d \frac{1}{n} k = 0 \sum n - 1 \frac{( k n - 1 ) ^{a} ( k - n - 1 ) ^{a}}{4 k ^{2} - 1} \in Z .$ We also solve a conjecture of V.J.W. Guo and J. Zeng, and pose several conjectures for further research.

32 pages

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New congruences for sums involving Apery numbers or central Delannoy numbers

Two new kinds of numbers and related divisibility results

Abstract

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