SOME RESULTS ON NAIVE TRANSCENDENCE IN THE RING OF INTEGERS MODULO INFINITELY LARGE PRIMES

Abstract

Abstract This paper presents various transcendence results in the ring of integers modulo infinitely large primes A $\mathcal {A}$ script upper A . In the ring A $\mathcal {A}$ script upper A , one can consider two notions of transcendence. One is based on the notion of finite algebraic numbers introduced by Rosen, while the other is transcendence in the naive sense. It is known that transcendence in the latter sense automatically implies transcendence in the former sense. In this paper, we strengthen results due to Anzawa and Funakura and to Luca and Zudilin by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. We also present several examples of naive transcendental numbers that do not seem to have appeared previously in the literature. Although we are not able to establish naive transcendence for certain numbers, we prove the irrationality of numbers such as log A ⁡ ( 2 ) $\log _{\mathcal {A}}(2)$ log Subscript script upper A Baseline left parenthesis 2 right parenthesis under the ABC conjecture.