Displaying similar documents to “On p -adic zeros of systems of diagonal forms restricted by a congruence condition”

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 · 509203 (k = 1, 2,...) is of the form n - φ(n).

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...

Modified proof of a local analogue of the Grothendieck conjecture

Victor Abrashkin (2010)

Journal de Théorie des Nombres de Bordeaux

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A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields K with finite residue fields of characteristic p ≠ 0 and the category of absolute Galois groups of fields K together with their ramification filtrations. The case of characteristic 0 fields K was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case p > 2 (but with no restrictions on the characteristic of K )....