Explicit Bounds For The Diophantine Equation A!B! = C!
arXiv:1903.08370
Abstract
A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A B C -- 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6. Several estimates on the relative size of the parameters are known, such as the one given by Erd{ö}s C -- B 5 log log C, or the one given by Bhat and Ramachandra C --B (1/ log 2+o(1)) log log C. We check the conjecture for B 10 3000 and give better explicit bounds such as C -- B log log(B+1) log 2 -- 0.8803.
arXiv commentThe Fibonacci Quarterly, Dalhousie University, In press