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A generalization of a curious congruence on harmonic sums

Zhao established a curious harmonic congruence for prime p > 3: Σ 1 ιjk≡-2B p-3 (mod p). i+j+k=p i,j,k>0 In this note the authors extend it to the following congruence for any prime p > 3 and… Expand

Some identities for multiple zeta values

- Zhongyan Shen, T. Cai
- Mathematics
- 1 February 2012

Abstract In this note, we obtain the following identities, ∑ a + b + c = n ζ ( 2 a , 2 b , 2 c ) = 5 8 ζ ( 2 n ) − 1 4 ζ ( 2 ) ζ ( 2 n − 2 ) , for n > 2 , ∑ a + b + c + d = n ζ ( 2 a , 2 b , 2 c , 2… Expand

N-tuples of Positive Integers with the Same Sum and the Same Product

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BERNOULLI NUMBERS AND CONGRUENCES FOR HARMONIC SUMS

- Binzhou Xia, T. Cai
- Mathematics
- 1 June 2010

Zhao established the following harmonic congruence for prime p > 3: In this note, the authors improve it to the following congruence for prime p > 5: Meanwhile, they also improve a generalization of… Expand

ON 2-NIVEN NUMBERS AND 3-NIVEN NUMBERS

- T. Cai
- Mathematics
- 1996

A Niven number [3] is a positive integer that is divisible by the sum of its digits. Various papers have appeared concerning digital sums and properties of the set of Niven numbers. In 1993, Cooper… Expand

On products of consecutive arithmetic progressions

Abstract In this paper, first, we show the Diophantine equation x ( x + b ) y ( y + b ) = z ( z + b ) has infinitely many nontrivial positive integer solutions for b ≥ 3 . Second, we prove the… Expand

A curious congruence modulo prime powers

- Liuquan Wang, T. Cai
- Mathematics
- 1 November 2014

Abstract Zhao established a curious congruence, i.e., for any prime p ≥ 3 , ∑ i + j + k = p i , j , k > 0 1 i j k ≡ − 2 B p − 3 ( mod p ) . In this note we prove that for any prime p ≥ 3 and positive… Expand

Super congruences involving alternating harmonic sums modulo prime powers

- Zhongyan Shen, T. Cai
- Mathematics
- 11 March 2015

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$, \begin{equation*}
\sum\limits_{i+j+k=p^{r}\atop{i,j,k\in… Expand

Perfect numbers and Fibonacci primes (III)

In this article, we consider the Diophantine equation $\sigma_{2}(n)-n^2=An+B$ with $A=P^2\pm2$. For some $B$, we show that except for finitely many computable solutions in the range… Expand

Figurate primes and Hilbert's 8th problem

- T. Cai, Y. Zhang, Zhongyan Shen
- Mathematics
- 22 June 2014

In this paper, by using the theory of elliptic curves, we discuss several Diophantine equations related with the so-called figurate primes. Meanwhile, we raise several conjectures related with… Expand

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