### Electronic Research Archive

2021, Issue 6: 3853-3865. doi: 10.3934/era.2021065

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

• Received: 01 May 2021 Revised: 01 July 2021 Published: 07 September 2021
• Primary: 11E41, 11R29; Secondary: 11E16

• 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 $\mathbb Q(\sqrt{-p}), \ p = 4n-1$[J]. Electronic Research Archive, 2021, 29(6): 3853-3865. doi: 10.3934/era.2021065

### Related Papers:

• 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. ###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142 1.833

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