Totient divisibility by $n^2+1$ and $n^2+2$
Year: 2026
Type: article
Abstract: This note establishes exact classifications of the positive integers (n) satisfying two divisibility conditions involving Euler's totient function: [\varphi(n)\mid n^2+1\quad\Longleftrightarrow\quadn\in{1,2,3},] and [\varphi(n)\mid n^2+2\quad\Longleftrightarrow\quadn\in{1,2,4,6,14}.] The proofs use elementary number-theoretic arguments involving parity, prime factorization, and (2)-adic valuations. A computational verification using an exact totient sieve is also provided, together with a Python script for reproducing the exhaustive checks up to (2\times10^7). The note additionally records computational observations for related divisibility conditions of the form (\varphi(n)\mid n^2+c), without claiming complete classifications for those cases.
Author: Freeman, James
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225650091
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