EXPLICIT BOUNDS FOR THE DIOPHANTINE EQUATION A!B! = C!

International audienceA 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 Erdös 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