Displaying similar documents to “Solutions of some classes of congruences”

On p -adic zeros of systems of diagonal forms restricted by a congruence condition

Hemar Godhino, Paulo H. A. Rodrigues (2007)

Journal de Théorie des Nombres de Bordeaux

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This paper is concerned with non-trivial solvability in p -adic integers of systems of additive forms. Assuming that the congruence equation a x k + b y k + c z k d ( m o d p ) has a solution with x y z 0 ( m o d p ) we have proved that any system of R additive forms of degree k with at least 2 · 3 R - 1 · k + 1 variables, has always non-trivial p -adic solutions, provided p k . The assumption of the solubility of the above congruence equation is guaranteed, for example, if p > k 4 .

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.