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Research PapersNo Access

A congruence involving harmonic sums modulo $p^{α} q^{β}$ $p^{α} q^{β}$

Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R. China

E-mail Address: txcai@zju.edu.cn

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,

Department of Mathematics, Zhejiang International Studies University, Hangzhou 310012, P.R. China

E-mail Address: huanchenszyan@163.com

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, and

Department of Mathematics, Zhejiang University, Hangzhou 310027, P.R. China

E-mail Address: jialirui@126.com

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https://doi.org/10.1142/S1793042117500580Cited by:3 (Source: Crossref)

Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime $p$ and positive integer $r$ ,

Z (p r) \equiv - 2 p r - 1 B p - 3 (mod p r), <math display="block" altimg="eq-00005.gif"><mrow><mi>Z</mi><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\equiv</mo><mo stretchy="false" class="MathClass-bin">-</mo><mn>2</mn><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi><mo stretchy="false" class="MathClass-bin">-</mo><mn>1</mn></mrow></msup><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo stretchy="false" class="MathClass-bin">-</mo><mn>3</mn></mrow></msub><mspace width="1em" class="quad"></mspace><mo class="MathClass-open" stretchy="false">(</mo><mstyle><mtext class="textrm" mathvariant="normal">mod</mtext></mstyle><mspace width=".17em" class="thinspace"></mspace><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo></mrow></math>

where

$Z (n) = \sum_{\binom{i + j + k = n}{i, j, k \in 𝒫_{n}}} \frac{1}{i j k}$ and

$𝒫_{n}$ denote the set of positive integers which are prime to

$n$ . In this paper, we obtain an unexpected congruence for distinct odd primes

$p$ ,

$q$ and positive integers

$α, β$ ,

Z(pαqβ)≡2(2−q)(1−1q3)pα−1qβ−1Bp−3(modpα)<math display="block" altimg="eq-00012.gif"><mrow><mi>Z</mi><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>β</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">≡</mo><mn>2</mn><mo class="MathClass-open" stretchy="false">(</mo><mn>2</mn><mo stretchy="false" class="MathClass-bin">−</mo><mi>q</mi><mo class="MathClass-close" stretchy="false">)</mo><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac></mrow></mfenced><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>β</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>3</mn></mrow></msub><mspace width="1em" class="quad"></mspace><mo class="MathClass-open" stretchy="false">(</mo><mstyle><mtext class="textrm" mathvariant="normal">mod</mtext></mstyle><mspace width=".17em" class="thinspace"></mspace><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo></mrow></math>

and the necessary and sufficient condition for

Z (p α q β) \equiv 0 (mod p α q β) . <math display="block" altimg="eq-00013.gif"><mrow><mi>Z</mi><mo class="MathClass-open" stretchy="false">(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>β</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\equiv</mo><mn>0</mn><mspace width="1em" class="quad"></mspace><mo class="MathClass-open" stretchy="false">(</mo><mstyle><mtext class="textrm" mathvariant="normal">mod</mtext></mstyle><mspace width=".17em" class="thinspace"></mspace><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>β</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">.</mo></mrow></math>

Finally, we raise a conjecture that for

$n > 1$ and odd prime power

$p^{α} ∥ n$ ,

$α \geq 1$ ,

Z(n)≡∏q|nq≠p(1−2q)(1−1q3)(−2np)Bp−3(modpα).<math display="block" altimg="eq-00017.gif"><mrow><mi>Z</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>n</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">≡</mo><munder class="msub"><mrow><mo class="MathClass-op">∏</mo></mrow><mrow><mfrac linethickness="0"><mrow><mi>q</mi><mo class="MathClass-rel">|</mo><mi>n</mi></mrow><mrow><mi>q</mi><mo class="MathClass-rel">≠</mo><mi>p</mi></mrow></mfrac></mrow></munder><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">−</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mi>q</mi></mrow></mfrac></mrow></mfenced><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo stretchy="false" class="MathClass-bin">−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac></mrow></mfenced><mfenced separators="" open="(" close=")"><mrow><mo stretchy="false" class="MathClass-bin">−</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></mfenced><msub><mrow><mi>B</mi></mrow><mrow><mi>p</mi><mo stretchy="false" class="MathClass-bin">−</mo><mn>3</mn></mrow></msub><mspace width="1em" class="quad"></mspace><mo class="MathClass-open" stretchy="false">(</mo><mstyle><mtext class="textrm" mathvariant="normal">mod</mtext></mstyle><mspace width=".17em" class="thinspace"></mspace><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">.</mo></mrow></math>

However, we fail to prove it even for

$n$ with three distinct prime factors.

Keywords:

AMSC: 11A07, 11A41

References

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Journal cover image

Vol. 13, No. 05

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Downloaded 62 times

History

Received 14 December 2015

Accepted 12 May 2016

Published: 7 December 2016

Keywords