Exponential triples

TL;DR

This paper proves that in any two-part partition of natural numbers, one cell contains infinitely many exponential triples, and introduces the concept of exponential IP sets related to multiplicative IP* sets.

Contribution

It establishes the existence of exponential triples in any bipartition of natural numbers and defines exponential IP sets as an analogue to IP sets under exponentiation.

Findings

Existence of infinitely many exponential triples in one cell of any bipartition of natural numbers.

Any multiplicative IP* set is also an exponential IP set.

Abstract

Using ultrafilter techniques we show that in any partition of $\mathbb{N}$ into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind $a,b,a^b$ (with $a,b>1$). Also, we will show that any multiplicative $IP^*$ set is an "exponential $IP$ set", the analogue of an $IP$ set with respect to exponentiation.