Lower limit for the number of sets of solutions of \(x^e + y^e + z^e\equiv 0(\text{mod.\,}p)\). (Q1489000)
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scientific article; zbMATH DE number 2636907
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Lower limit for the number of sets of solutions of \(x^e + y^e + z^e\equiv 0(\text{mod.\,}p)\). |
scientific article; zbMATH DE number 2636907 |
Statements
Lower limit for the number of sets of solutions of \(x^e + y^e + z^e\equiv 0(\text{mod.\,}p)\). (English)
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1909
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Der Verf. beweist: Sind \(e\) und \(p\) ungerade Primzahlen, so hat die Kongruenz \[ x^e+ y^e + z^e \equiv 0 (\text{mod.\,}p) \] stets ganzzahlige, zu \(p\) prime Lösung \(x,y,z\), wenn: \[ p\geq (e -1)^2 (e -2)^2 + 6e-2. \] Mit derselben Methode behandelt er den Fall \(e = 4\) und zeigt, daß für \(p = 4n + 1 > 17\) stets zu \(p\) teilerfremde Lösungen existieren.
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congruences
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Fermat equation
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