Gauss - Dirichlet class number formula

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Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
$$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






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    Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
    $$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






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      Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
      $$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






      nt.number-theory analytic-number-theory algebraic-number-theory







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      Let $p=8k+3$ be a prime. Then the class number of the imaginary quadratic field $mathbb Q(sqrt-p)$ is given by
      $$h(-p)=frac 13sum_k=1^fracp-12left(frac kp right).$$ While this is certainly a very elegant and compact formula, does it have any nontrivial applications or consequences? (Note that I am not asking about the Dirichlet class number formula relating the value of the corresponding $L$-series at $1$ with the class number, but rather about its particular corollary).






      nt.number-theory analytic-number-theory algebraic-number-theory




      nt.number-theory analytic-number-theory algebraic-number-theory






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      Shimrod

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          Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






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            Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






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              Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






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                Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.






                share|cite|improve this answer












                Let $p=2n+1$ be a prime with $pequiv3pmod4$. By Wilson's theorem, $ (n!)^2equiv1pmod p$ and hence $n!equivpm1pmod p$. L. J. Mordell [Amer. Math. Monthly 68(1961), 145-146] used Dirichlet's class number formula $$left(2-left(frac 2pright)right) h(-p)=sum_k=1^nleft(frac kpright)$$ to deduce in few lines that $n!equiv(-1)^(h(-p)+1)/2pmod p$ if $p>3$. This is a nice application of Dirichlet's class number formula.







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                answered 3 hours ago









                Zhi-Wei Sun

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