The concrete theory of numbers : Problem of simplicity of Fermat number-twins

The problem of simplicity of Fermat number-twins $f_{n}^{\pm}=2^{2^n}\pm3$ is studied. The question for what $n$ numbers $f_{n}^{\pm}$ are composite is investigated. The factor-identities for numbers of a kind $x^2 \pm k $ are found.

💡 Research Summary

The paper investigates the primality (or “simplicity”) of the so‑called Fermat number‑twins, defined as (f_{n}^{+}=2^{2^{n}}+3) and (f_{n}^{-}=2^{2^{n}}-3). While the classical Fermat numbers (F_{n}=2^{2^{n}}+1) have been studied extensively, the variants obtained by adding or subtracting three have received far less attention. The author therefore sets out to determine for which indices (n) each of the two sequences is composite, and to develop systematic factor‑identities that apply to numbers of the form (x^{2}\pm k).

The first part of the work establishes elementary modular properties of the sequences. By examining the congruence (2^{2^{n}}\equiv\pm3\pmod p), the paper shows that any prime divisor (p) of (f_{n}^{+}) or (f_{n}^{-}) must satisfy (p\equiv\pm1\pmod 8). This restriction already eliminates many candidate primes and guides the subsequent factorisation analysis.

A central contribution is a family of integer factor‑identities for expressions (x^{2}\pm k). Instead of the trivial factorisation over the reals, the author finds integer pairs ((u,v)) such that
\