ABC Conjecture: ABC = 2^m pⁿ q^r with Fermat or Mersenne Primes

2018/09/06 by Saxena, Anupam
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1809.03328

Abstract

For p and q any two distinct Fermat or Mersenne primes, m,n,r as positive integers and μ= ± 1 satisfying any diophantine relation, (i) 2^m + μ= pⁿq^r, (ii) 2^mpⁿ + μ= q^r or (iii) pⁿ + μq^r = 2^m, it is shown that the number of triplets \A, B, C \ with gcd(A,B) = 1 and C = A + B, for which their product is of the form ABC = 2^mpⁿq^r and which satisfy C > rad(ABC)^{1 + ε} for any real ε > 0, is finite. For the triplet \2^y+1, 2^2y+1, (2^y+1)²\, a solution to (iii) with positive integer y such that 2^y+1 and 2^2y+1 are primes, rad(ABC)^{1 + ε} > C holds for any ε > 0. Furthermore, finiteness of the number of solutions of (iii) when n is even, is demonstrated elsewhere (Ref. [64]). All other solutions are enumerated.

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