TRANSCENDENTAL PARAMETERS OF ANALOGS OF SQUARES FOR SOLVING A CERTAIN SYSTEM OF EQUATIONS (Analytic Number Theory and Related Topics)

Abstract

Let 𝘚(α) = {⎿α𝓃²⏌:𝓃=1, 2, ... } ∖ {0}. Kanado and the author showed that for all rational numbers α∈ (0, 1) we can find infinitely many tuples (𝓴, 𝓵, 𝓶) of positive integers such that all of 𝓴, 𝓵, 𝓶, 𝓴+𝓵, 𝓵+𝓶, 𝓶+𝓴, 𝓴+𝓵+𝓶 are in 𝘚(α). In this short article, we construct a transcendental number α∈(0, 1) satisfying this relation.