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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 andExpand
Some identities for multiple zeta values
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 , 2Expand
N-tuples of Positive Integers with the Same Sum and the Same Product
TLDR
In this paper, by using the theory of elliptic curves, we prove that for every k, there exists infinitely many primitive sets of n-tuples of positive integers with the same sum and product. Expand
BERNOULLI NUMBERS AND CONGRUENCES FOR HARMONIC SUMS
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 ofExpand
ON 2-NIVEN NUMBERS AND 3-NIVEN NUMBERS
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, CooperExpand
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 theExpand
A curious congruence modulo prime powers
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 positiveExpand
Super congruences involving alternating harmonic sums modulo prime powers
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\inExpand
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 rangeExpand
Figurate primes and Hilbert's 8th problem
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 withExpand
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