Transcendence of Montgomery Reduction Factor in the Ring of Integers Modulo Infinitely Large Primes

We prove the transcendence over $\mathbb{Q}$ of the image of Montgomery reduction factor $R^{\mathrm{\prime }}$ in the ring $\mathcal{A}$ of integers modulo infinitely large primes. Here, for an odd prime number $p$, $R^{\mathrm{\prime }}$ is defined as the modular inverse of a power $R$ of $2$ modulo $p$ satisfying $2^{-k}R\le p<R$ for a fixed constant $k\in {\mathbb{N}}_{>0}$ typically given as the standard bit size $32$ of an integer type, and is the element of $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ representing Montgomery reduction regarded as a $\mathbb{Z}\mathrm{/}p\mathbb{Z}$-linear homomorphism.