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Transcendence of Montgomery Reduction Factor in the Ring of Integers Modulo Infinitely Large Primes
Abstract
We prove the transcendence over ℚ of the image of Montgomery reduction factor R' in the ring 𝒜 of integers modulo infinitely large primes. Here, for an odd prime number p, R' is defined as the modular inverse of a power R of $2$ modulo p satisfying 2−k R ≤ p < R for a fixed constant k ∈ ℕ> 0 typically given as the standard bit size $32$ of an integer type, and is the element of ℤ/p ℤ representing Montgomery reduction regarded as a ℤ/p ℤ-linear homomorphism.
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BibTeX
@misc{mihara2026transcendence,
title = {{Transcendence of Montgomery Reduction Factor in the Ring of Integers Modulo Infinitely Large Primes}},
author = {Tomoki Mihara},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14822},
eprint = {2609.14822},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14822},
}
RIS
TY - UNPB
TI - Transcendence of Montgomery Reduction Factor in the Ring of Integers Modulo Infinitely Large Primes
AU - Tomoki Mihara
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14822
UR - https://arxiv.org/abs/2609.14822
AB - We prove the transcendence over ℚ of the image of Montgomery reduction factor R' in the ring 𝒜 of integers modulo infinitely large primes. Here, for an odd prime number p, R' is defined as the modular inverse of a power R of $2$ modulo p satisfying 2^−k R ≤ p < R for a fixed constant k ∈ ℕ_> 0 typically given as the standard bit size $32$ of an integer type, and is the element of ℤ/p ℤ representing Montgomery reduction regarded as a ℤ/p ℤ-linear homomorphism.
ER -