On the diophantine equation $y^2 + D^m = p^n$

Masao Toyoizumi

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A class of diophantine equations connected with certain elliptic curves over $Q (\sqrt{- 13})$

R. J. Stroeker (1979)

Compositio Mathematica

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The conductor of a cyclic quartic field using Gauss sums

Blair K. Spearman, Kenneth S. Williams (1997)

Czechoslovak Mathematical Journal

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Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$ . It is known that there are unique integers $A$ , $B$ , $C$ , $D$ such that $K = Q (\sqrt{A (D + B \sqrt{D})}),$ where $A is squarefree and odd, D = B^{2} + C^{2} is squarefree, B > 0, C > 0, G C D (A, D) = 1 .$ The conductor $f (K)$ of $K$ is $f (K) = 2^{l} | A | D$ , where $l = \{\begin{matrix} 3, if D \equiv 2 (mod 4) or D \equiv 1 (mod 4), B \equiv 1 (mod 2), \\ 2, if D \equiv 1 (mod 4), B \equiv 0 (mod 2), A + B \equiv 3 (mod 4), \\ 0, if D \equiv 1 (mod 4), B \equiv 0 (mod 2), A + B \equiv 1 (mod 4) . \end{matrix}$ A simple proof of this formula for $f (K)$ is given, which uses the basic properties of quartic Gauss sums.

On the class numbers of Q(√±2p) modulo 16, for p ≡ 1 (mod 8) a prime

Pierre Kaplan, Kenneth Williams (1982)

Acta Arithmetica

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On the parity of the class number of the field $ℚ (\sqrt{p}, \sqrt{q}, \sqrt{r})$

Michal Bulant (1998)

Acta Mathematica et Informatica Universitatis Ostraviensis

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Displaying similar documents to “On the diophantine equation y 2 + D m = p n ”

A class of diophantine equations connected with certain elliptic curves over Q ( - 13 )

The conductor of a cyclic quartic field using Gauss sums

On the class numbers of Q(√±2p) modulo 16, for p ≡ 1 (mod 8) a prime

On the parity of the class number of the field ℚ ( p , q , r )

Displaying similar documents to “On the diophantine equation $y^{2} + D^{m} = p^{n}$ ”

A class of diophantine equations connected with certain elliptic curves over $Q (\sqrt{- 13})$

On the parity of the class number of the field $ℚ (\sqrt{p}, \sqrt{q}, \sqrt{r})$