On Euler's "Misleading Induction", Andrews' "Fix", and How to Fully Automate them

Abstract

One of the greatest experimental mathematicians of all time was also one of the greatest mathematicians of all time, the great Leonhard Euler. Usually he had an uncanny intuition on how many "special cases" one needs before one can formulate a plausible conjecture, but one time he was "almost fooled", only to find out that his conjecture was premature. In 1990, George Andrews found a way to "correct" Euler. Here we show how to generate, AUTOMATICALLY, rigorously-proved Euler-Andrews Style formulas, that enables one to generate Euler-style "cautionary tales" about the "danger" of using naive empirical induction. Ironically, the way we prove the Andrews-style corrections is empirical! But in order to turn the empirical proof into a full-fledged rigorous proof, we must make sure that we check sufficiently many (but still not that many!) special cases.