2 Hao Pan and Zhi

. N T ] 1 0 Ju l 2 00 6 MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS ( II )

For x ∈ Z let T x denote the triangular number x(x + 1)/2. Following the recent approach of Z. W. Sun, we show that every natural number can be written in any of the following forms with x, y, z ∈ Z:

Mixed Sums of Squares and Triangular Numbers

For x ∈ Z let Tx denote the triangular number x(x + 1)/2. Following the recent approach of Z. W. Sun, we show that every natural number can be written in any of the following forms with x, y, z ∈ Z: x + Ty + Tz , x 2 + 2Ty + Tz , x 2 + 3Ty + Tz , x + 5Ty + 2Tz , x 2 + 6Ty + Tz , 3x 2 + 2Ty + Tz , x + 3y + Tz , 2Tx + Ty + Tz , 3Tx + 2Ty + Tz , 5Tx + Ty + Tz . This confirms some conjectures raise...

An Identity Involving Products of Three Binomial Coefficients

for any a, b, c ∈ N = {0, 1, 2, . . .}. During the second author’s visit (January–March, 2005) to the Institute of Camille Jordan at Univ. Lyon-I, Dr. Victor Jun Wei Guo told Sun his following conjecture involving sums of products of three binomial coefficients and said that he failed to prove this “difficult conjecture” during the past two years though he had tried to work it out again and again.

Preprint, arXiv:1010.2489 PROOF OF THREE CONJECTURES ON CONGRUENCES

In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let p be an odd prime and let a be a positive integer. We show that if p ≡ 1 (mod 4) or a > 1 then ⌊ 3 4 p a ⌋ k=0 −1/2 k ≡ 2 p a (mod p 2),. where (−) denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing...

ON q - EULER NUMBERS AND q - SALIÉ