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001-es BibID:BIBFORM008913
Első szerző:Győry Kálmán (matematikus)
Cím:Perfect powers from products of consecitive terms in arithmetic progression / K. Győry, L. Hajdu, Á. Pintér
Megjegyzések:Abstract: We prove that for any positive integers x, d, k with gcd(x, d) = 1 and 3 < k < 35 the product x(x + d) . . . (x + (k - 1)d) cannot be a perfect power. This yields a considerable extension of previous results of Győry, Hajdu, Saradha, and Bennett, Bruin, Győry, Hajdu which covered the cases k <= 11.We also establish more general theorems for the case when x can also be a negative integer and the product yields an almost perfect power. As in the proofs of the earlier theorems, for fixed k we reduce the problem to systems of ternary equations. However, our results do not follow as a mere computational sharpening of the approach utilized previously, but instead require the introduction of fundamentally new ideas. For k > 11, a great number of new ternary equations arise that we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the different cases in the usual manner. The main novelty of this paper is that we algorithmize our proofs which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the arising new ternary equations with several sieves based on the ternary equations already solved. In this way we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger k as well, but there is of course a computational time limit.
Tárgyszavak:Természettudományok Matematika- és számítástudományok idegen nyelvű folyóiratközlemény külföldi lapban
diofantikus egyenletek
Megjelenés:Compositio Mathematica. - 145 : 4 (2009), p. 845-864. -
További szerzők:Hajdu Lajos (1968-) (matematikus) Pintér Ákos (1967-) (matematikus)
Internet cím:elektronikus változat
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