Abstract
For a homogenous polynomP ∈ℤ[X, Y] of degreed and forh ∈ ℕ letL(P, h) be the number of coprime solutions of the equation |P(x,y)|=h. Ift(h) is the number of distinct primefactors ofh, a theorem of Bombieri-Schmidt [1] givesL(P, h)≤Md t(h) +1 in the cased≥3. We prove for a finite collection of polynomialsP w ∈ℤ[X, Y] under some conditions, that\(\mathop \sum \limits_{w = 1}^k L(P_w ,h) \leqslant 2d^{t(h)} \) for almost allh ∈ℤ (Satz 1; “almost all” in the sense “except finitely many cases”). As a corollary (Folgerung 3/4) we get for sufficiently large primesp, that the equation |x d−c z y d|=p has at mostd+1 many solutions (x, y, z) ∈ ℕ 3 withc∤y. Ford=2 we get an analogon to a theorem of Mao-Hua (Folgerung 5).
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Langmann, K. Lösungsanzahl der Gleichung |xd−czyd|=p. Manuscripta Math 84, 389–400 (1994). https://doi.org/10.1007/BF02567464
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DOI: https://doi.org/10.1007/BF02567464