Abstract
For a positive integer let be the th harmonic number. In this paper we prove that, for any prime , . Notice that the first part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers.
Highlights
Introduction and Main ResultsGiven positive integers nn and mm, the harmonic numbers of order mm are those rational numbers HHnnnnn de ned as HHnnnnn = nn kkkk 1 kkmm (1)For simplicity, we will denote by HHnn ∶=HHnnnn kkkk kk (2)the nnth harmonic number.Usually, here as always in the sequel, we consider the congruence relation modulo a prime pp extended to the ring of rational numbers with denominators not divisible by pp
We will denote by the nnth harmonic number
Here as always in the sequel, we consider the congruence relation modulo a prime pp extended to the ring of rational numbers with denominators not divisible by pp. For such fractions we put mmmmmm mmmmm mmmmmmm if and only if mmmm mmmmm mmmmmmm, and the residue class of mmmmm is the residue class of mmmmm where nnn is the inverse of nn modulo pp
Summary
For a positive integer nn let HHnn = ∑nknkkk 1/nn be the nnth harmonic number. In this paper we prove that, for any prime pp p 7, ∑pkkpkpkp HHkk/kk2 ≡ ∑pkkpkpkp HH2kk/kk k kkkkkkk kpkkpkpkp 1/kk2(mod pp). Notice that the rst part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers
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