Powers and alternative laws

Nicholas Ormes; Petr Vojtěchovský

Displaying similar documents to “Powers and alternative laws”

A note on a conjecture of Jeśmanowicz

Moujie Deng, G. Cohen (2000)

Colloquium Mathematicae

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Let a, b, c be relatively prime positive integers such that $a^{2} + b^{2} = c^{2}$ . Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ${(a n)}^{x} + {(b n)}^{y} = {(c n)}^{z}$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ${(u^{2} - v^{2})}^{x} + {(2 u v)}^{y} = {(u^{2} + v^{2})}^{z}$ , for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.

Two exponential diophantine equations

Dominik J. Leitner (2011)

Journal de Théorie des Nombres de Bordeaux

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The equation $3^{a} + 5^{b} - 7^{c} = 1$ , to be solved in non-negative rational integers $a, b, c$ , has been mentioned by Masser as an example for which there is still no algorithm to solve completely. Despite this, we find here all the solutions. The equation $y^{2} = 3^{a} + 2^{b} + 1$ , to be solved in non-negative rational integers $a, b$ and a rational integer $y$ , has been mentioned by Corvaja and Zannier as an example for which the number of solutions is not yet known even to be finite. But we find here all the solutions too; there are in fact...

On integers not of the form n - φ (n)

J. Browkin, A. Schinzel (1995)

Colloquium Mathematicae

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W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers $2^{k} \cdot 509203$ (k = 1, 2,...) is of the form n - φ(n).