On the Diophantine Equation
نویسنده
چکیده
In this paper, we study the Diophantine equation x2 + C = 2yn in positive integers x, y with gcd(x, y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4) we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequence due Bilu, Hanrot and Voutier. When C 6≡ 1 (mod 4) we explain how the equation can be solved using the multi-Frey variant of the modular approach. We illustrate our approach by solving completely the equations x2 + 17a1 = 2yn, x2 + 5a113a2 = 2yn, and x2 + 3a111a2 = 2yn.
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