2022/11/24 by LE Mao-hua, Reese Scott, Le, Maohua +3
Mathematics · #11D61 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2211.13378
openalex publication_date 2022/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let a, b, c be distinct primes with a<b>1 and (a,b,c) is not one of the six known such triples then (a,b,c) must be one of three cases. In the present paper, we eliminate two of these cases (using the special properties of certain continued fractions for one of these cases, and using a result of Dirichlet on quartic residues for the other). Then we show that the single remaining case requires severe restrictions, including the following: a=2, b ≡ 1 \bmod 48, c ≡ 17 \mod 48, b > 109, c > 1018; at least one of the multiplicative orders uc(b) or ub(c) must be odd (where up(n) is the least integer t such that nt ≡ 1 \bmod p); 2 must be an octic residue modulo c except for one specific case; 2 | v2(b-1) ≤ v2(c-1) (where v2(n) satisfies 2v2(n) ∥ n); there must be exactly two solutions (x1, y1, z1) and (x2, y2, z2) with 1 = z1 < z2 and either x1 ≥ 28 or x2 ≥ 88. These results support a conjecture put forward in \citeScSt6 and improve results in \citeLeSt.</b>