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failure function (Diophantine equations) (Definition)

Let $ x^{2} + c = a^{n }$ be a Diophantine equation with $ x,a,c,n\in \mathbb{N}$, $ a$ and $ c$ fixed. Let the definition of a failure be a non-solution of this equation. Let $ (x_0)^{2} + c =a^{s}\times l$ for $ l\in \mathbb{N}$, $ l\neq 1$ and $ x_0$ a fixed value of $ x$. Then $ x= \psi( x_0) = x_0 + k\times a^{s}$ is a failure function $ (k \in\mathbb{N}.$ [1].

1. "A Theorem a la Ramanujan".



"failure function (Diophantine equations)" is owned by akdevaraj. [ full author list (2) ]
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Cross-references: Ramanujan, function, equation, fixed, Diophantine equation

This is version 7 of failure function (Diophantine equations), born on 2008-04-28, modified 2008-04-28.
Object id is 10550, canonical name is FailureFunctionDiophantineEquations.
Accessed 230 times total.

Classification:
AMS MSC: 33-01 (Special functions :: Instructional exposition )
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