On the mod $p^2$ determination of $\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)$: another proof of a conjecture by Sun

Details On the mod $p^2$ determination of $\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)$: another proof of a conjecture by Sun On the mod $p^2$ determination of $\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)$: another proof of a conjecture by Sun

2011-08-16

arxiv.org/abs/1108.3197v2

Mathematics

Number Theory

Pages 20

Scientific paper

For a positive integer $n$ let $H_n=\sum_{k=1}^{n}1/k$ be the $n$th harmonic number. Z. W. Sun conjectured that for any prime $p\ge 5$, $$ \sum_{k=1}^{p-1}\frac{H_k}{k\cdot 2^k} \equiv7/24pB_{p-3}\pmod{p^2}. $$ This conjecture is recently confirmed by Z. W. Sun and L. L. Zhao. In this note we give another proof of the above congruence by establishing congruences for all the sums of the form $\sum_{k=1}^{p-1}2^{\pm k}H_k^r/k^s \,(\bmod{\, p^{4-r-s}})$ with $(r,s)\in\{(1,1),(1,2),(2,1) \}$.

Affiliated with

Also associated with

No associations

Rating

On the mod $p^2$ determination of $\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)$: another proof of a conjecture by Sun does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the mod $p^2$ determination of $\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)$: another proof of a conjecture by Sun, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the mod $p^2$ determination of $\sum_{k=1}^{p-1}H_k/(k\cdot 2^k)$: another proof of a conjecture by Sun will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-195222