Year: 2026
Type: preprint
Abstract: We provide progress on the characterization of composite integers $n$ that satisfy the jump conditions $φ(n+12)=φ(n)+12$ and $σ(n+12)=σ(n)+12$ simultaneously. While it is known that prime quadruplets $(p,p+2,p+6,p+8)$ generate solutions $n=p(p+8)$, the complete characterization remains an open conjecture. We prove that this characterization is complete when $n$ and $n+12$ are both squarefree semiprimes, and that no solution $n$ can be a prime power. Furthermore, a complete search up to $10^{12}$ resulted in no counterexamples to the conjecture. If this conjecture is proven true, and there are infinitely many such solutions, then it can be proved that there are infinitely many prime quadruplets.
Source: arXiv (Cornell University)
Authors: Choudhury, Himaghna Roy, Wei, Shicheng
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164140346