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- W2899178246 abstract "Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mrow> <m:mrow> <m:mrow> <m:munder> <m:mo largeop=true movablelimits=false symmetric=true>∑</m:mo> <m:mtable rowspacing=0.0pt> <m:mtr> <m:mtd> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>+</m:mo> <m:mi>j</m:mi> <m:mo>+</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>p</m:mi> <m:mi>r</m:mi> </m:msup> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>j</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi mathvariant=script>P</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:munder> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:mi>j</m:mi> <m:mo></m:mo> <m:mi>k</m:mi> </m:mrow> </m:mfrac> </m:mrow> <m:mo>≡</m:mo> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:msup> <m:mi>p</m:mi> <m:mrow> <m:mi>r</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>B</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mrow> <m:mspace width=veryverythickmathspace /> <m:mrow> <m:mo lspace=8.1pt stretchy=false>(</m:mo> <m:mrow> <m:mo movablelimits=false>mod</m:mo> <m:msup> <m:mi>p</m:mi> <m:mi>r</m:mi> </m:msup> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> $$sum_{begin{subarray}{c}i+j+k=p^{r} i,j,kinmathcal{P}_{p}end{subarray}}frac{1}{ijk}equiv-2p^{r-1}B_{p-3} quadquad(text{mod} ,, {p^{r}}),$$ where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msub> <m:mi mathvariant=script>P</m:mi> <m:mi>n</m:mi> </m:msub> </m:math> $ mathcal{P}_{n} $ denote the set of positive integers which are prime to n . In this note, we obtain the congruences for distinct odd primes p , q and positive integers α , β , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mrow> <m:mrow> <m:munder> <m:mo largeop=true movablelimits=false symmetric=true>∑</m:mo> <m:mtable rowspacing=0.0pt> <m:mtr> <m:mtd> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>+</m:mo> <m:mi>j</m:mi> <m:mo>+</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>p</m:mi> <m:mi>α</m:mi> </m:msup> <m:mo></m:mo> <m:msup> <m:mi>q</m:mi> <m:mi>β</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>j</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi mathvariant=script>P</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>p</m:mi> <m:mo></m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:munder> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:mi>j</m:mi> <m:mo></m:mo> <m:mi>k</m:mi> </m:mrow> </m:mfrac> </m:mrow> <m:mo>≡</m:mo> <m:mrow> <m:mrow> <m:mfrac> <m:mn>7</m:mn> <m:mn>8</m:mn> </m:mfrac> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>-</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:msup> <m:mi>q</m:mi> <m:mn>3</m:mn> </m:msup> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:msup> <m:mi>p</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:msup> <m:mi>q</m:mi> <m:mrow> <m:mi>β</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>B</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mspace width=veryverythickmathspace /> <m:mrow> <m:mo lspace=8.1pt stretchy=false>(</m:mo> <m:mrow> <m:mo movablelimits=false>mod</m:mo> <m:msup> <m:mi>p</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> $$ sum_{begin{subarray}{c}i+j+k=p^{alpha}q^{beta} i,j,kinmathcal{P}_{2pq}end{subarray}}frac{1}{ijk}equivfrac{7}{8}left(2-% qright)left(1-frac{1}{q^{3}}right)p^{alpha-1}q^{beta-1}B_{p-3}pmod{p^{% alpha}} $$ and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mrow> <m:mrow> <m:mrow> <m:munder> <m:mo largeop=true movablelimits=false symmetric=true>∑</m:mo> <m:mtable rowspacing=0.0pt> <m:mtr> <m:mtd> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>+</m:mo> <m:mi>j</m:mi> <m:mo>+</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>p</m:mi> <m:mi>α</m:mi> </m:msup> <m:mo></m:mo> <m:msup> <m:mi>q</m:mi> <m:mi>β</m:mi> </m:msup> </m:mrow> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd> <m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>j</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:msub> <m:mi mathvariant=script>P</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo></m:mo> <m:mi>q</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:munder> <m:mfrac> <m:msup> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mi>i</m:mi> </m:msup> <m:mrow> <m:mi>i</m:mi> <m:mo></m:mo> <m:mi>j</m:mi> <m:mo></m:mo> <m:mi>k</m:mi> </m:mrow> </m:mfrac> </m:mrow> <m:mo>≡</m:mo> <m:mrow> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:msup> <m:mi>q</m:mi> <m:mn>3</m:mn> </m:msup> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo></m:mo> <m:msup> <m:mi>p</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:msup> <m:mi>q</m:mi> <m:mrow> <m:mi>β</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo></m:mo> <m:msub> <m:mi>B</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mspace width=veryverythickmathspace /> <m:mrow> <m:mo lspace=8.1pt stretchy=false>(</m:mo> <m:mrow> <m:mo movablelimits=false>mod</m:mo> <m:msup> <m:mi>p</m:mi> <m:mi>α</m:mi> </m:msup> </m:mrow> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> $$ sum_{begin{subarray}{c}i+j+k=p^{alpha}q^{beta} i,j,kinmathcal{P}_{pq}end{subarray}}frac{(-1)^{i}}{ijk}equivfrac{1}{2}% left(q-2right)left(1-frac{1}{q^{3}}right)p^{alpha-1}q^{beta-1}B_{p-3}% pmod{p^{alpha}}. $$" @default.
- W2899178246 created "2018-11-09" @default.
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- W2899178246 date "2018-10-01" @default.
- W2899178246 modified "2023-09-27" @default.
- W2899178246 title "Congruences involving alternating harmonic sums modulo p α q β" @default.
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