A REVIEW ON NON-UNIQUE REPRESENTATIONS OF POSITIVE INTEGERS AS 2X + 3Y FOR NON-NEGATIVE INTEGERS X AND Y

Abstract

We have reviewed the proof[1] of a conjecture posted in the Online Encyclopedia of Integer Sequences (OEIS)[2], which states that there are exactly five positive integers that can be represented in more than one way as the sum of non-negative integral powers of 2 and 3. The case for both powers being positive follows from a theorem of Bennett. We have also devised alternative logics to the elementary methods applied in the original paper “On duplicate representations as 2x+3y for nonnegative integers x and y”[1] to prove the case where zero exponents are allowed. Besides, we have included a nominal program towards the end, that verifies the statement of the conjecture.