End behavior of Ramanujan’s Taxicab numbers

Abstract Generalized taxicab numbers are the smallest positive integers that are the sum of exactly j , positive $$k^{\textrm{th}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>k</mml:mi> <mml:mtext>th</mml:mtext> </mml:msup> </mml:math> powers in exactly m distinct ways. This paper considers for which values of m does the smallest such positive integer exist as j gets large. There appear to be only two possible outcomes, leading to curious results, for instance, there is no positive integer that can be expressed as the sum of exactly 10 perfect squares in exactly 3 ways. This paper resolves a number of conjectures found in the OEIS by considering generalized Taxicab numbers in the setting of the theory of partitions.