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December  2021, 29(6): 3853-3865. doi: 10.3934/era.2021065

A computable formula for the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}), \ p = 4n-1$

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Received  May 2021 Revised  July 2021 Published  December 2021 Early access  September 2021

Using elementary methods, we count the quadratic residues of a prime number of the form $p = 4n-1$ in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number $h$ of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$ Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.

Citation: Jorge Garcia Villeda. A computable formula for the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}), \ p = 4n-1$. Electronic Research Archive, 2021, 29 (6) : 3853-3865. doi: 10.3934/era.2021065
References:
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References:
 [1] H. Cohen, A Course in Computational Algebraic Number Theory, Volume 138 of Graduate Text in Mathematics, Springer-Verlag, New York, 1993. doi: 10.1007/978-3-662-02945-9.  Google Scholar [2] L. E. Dickson, Introduction to the Theory of Numbers, Dover Publ. Inc., New York, 1957. Google Scholar [3] P. G. L. Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Volume 1, Cambridge University Press, (2012), 313–342. doi: 10.1017/CBO9781139237338.023.  Google Scholar [4] J. Garcia, Sum of quadratic-type residues modulus a prime $p = 4n-1$, work in progress. Google Scholar [5] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer, 2004. doi: 10.1007/978-3-662-07001-7.  Google Scholar [6] D. Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Volume XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., (1971), 415–440.  Google Scholar
The resulting case when $n-1\in (R_{M-1},Q_{M-1}]$
The resulting case when $n-1\in [R_M,Q_M)$
The quadratic residues $r = {r_p}\left( {{n^2}} \right)$ for each of the four different cases of $n$
 $n$ $M$ $p$ $r$ $4k$ $k$ $16k-1$ $M$ $4k+1$ $k$ $16k+3$ $5M+1$ $4k+2$ $k$ $16k+7$ $9M+4$ $4k+3$ $k$ $16k+11$ $13M+9$
 $n$ $M$ $p$ $r$ $4k$ $k$ $16k-1$ $M$ $4k+1$ $k$ $16k+3$ $5M+1$ $4k+2$ $k$ $16k+7$ $9M+4$ $4k+3$ $k$ $16k+11$ $13M+9$
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