Nonlinearity In Similar Structures: On (3a−1)(3b−1) = (5c−1)(5d−1), And gu=fv.

Abstract

Liptai, Nemeth, et. al. (2020) conjectured (and supposedly proved) that in the diophantine equation (3a−1)(3b−1)=(5c−1)(5d−1) in positive integers a≤b, and c≤d, the only solution to the title equation is (a,b,c,d)=(1,2,1,1). This article proves that the Liptai, Nemeth, et. al. (2020) conjecture and results are wrong, and that there is more than one solution for the equation (3a−1)(3b−1)=(5c−1)(5d−1). This article introduces “Existence Conditions” and new theories of “Rational Equivalence”, and a new theorem pertaining to the equation gu=fv.