Some results on naive transcendence in the ring of integers modulo infinitely large primes

TL;DR

This paper explores naive transcendence in the ring of integers modulo infinitely large primes, strengthening existing results and providing new examples of naive transcendental numbers.

Contribution

It improves previous theorems by removing assumptions and introduces new naive transcendental numbers, also linking irrationality to the ABC conjecture.

Findings

Strengthened results on naive transcendence by removing assumptions.

Constructed new examples of naive transcendental numbers.

Proved irrationality of certain numbers assuming the ABC conjecture.

Abstract

This paper presents various transcendence results in the ring of integers modulo infinitely large primes $\mathcal{A}$. In the ring $\mathcal{A}$, 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 of Anzawa-Funakura and Luca-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 $\log_{\mathcal{A}}(2)$ under the ABC conjecture.