The concrete theory of numbers : New Mersenne conjectures. Simplicity and other wonderful properties of numbers $L(n) = 2^{2n}pm2^npm1$

New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n) = 2^{2n}\pm2^n\pm1$ are investigated. Wonderful formulas $gcd $ for numbers $L (n) $ and numbers repunit are proved.

💡 Research Summary

The paper under review introduces four families of integers defined by the formula
(L_1(n)=2^{2n}+2^{n}+1,; L_2(n)=2^{2n}+2^{n}-1,; L_3(n)=2^{2n}-2^{n}+1,; L_4(n)=2^{2n}-2^{n}-1)
and calls the study “New Mersenne conjectures”. The author’s aim is to investigate three classical questions for each family: (i) when the numbers are prime, (ii) what common divisors two members may share, and (iii) whether the numbers are free of square factors. In addition, a set of “wonderful” greatest‑common‑divisor (gcd) formulas linking the (L_i) sequences to generalized repunits is presented.

Prime‑value observations
For (L_1) the author lists only three known prime values: (L_1(1)=7), (L_1(3)=73) and (L_1(9)=262657). He conjectures that primes can occur only when the exponent (n) is a power of three with an odd exponent (i.e. (n=3^{k}) with (k) odd). The argument rests on the elementary congruences (2\equiv-1\pmod 3) and (2^{3}\equiv1\pmod 7); no deeper analytic or algebraic proof is offered. Similar empirical tables are given for the other three families: a handful of small‑(n) primes for (L_2) (e.g. (5,19,71,271,4159)), a few primes for (L_3) (e.g. (13,241,18446744069414584321)), and a short list of “prime twins” for (L_4). The paper does not address density, infinitude, or any heuristic model; the prime‑value part is essentially a collection of computational data.

GCD “isolation” results
The core theoretical contribution consists of two families of identities. For the first family (Theorem 1) the author claims
\