Perfect powers in sequences of polygonal numbers
Journal of Number Theory, vol.291, pp.74-100, 2027 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 291
- Publication Date: 2027
- Doi Number: 10.1016/j.jnt.2026.04.004
- Journal Name: Journal of Number Theory
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO), Engineering Source (EBSCO)
- Page Numbers: pp.74-100
- Keywords: Baker's method, Binomial Thue equations, Computer solution of Diophantine equations, Diophantine equations, Generalised Riemann hypothesis, Polygonal numbers, Weak effective abc conjecture
- Open Archive Collection: AVESIS Open Access Collection
- Bursa Uludag University Affiliated: Yes
Abstract
Let Ps(n) denote the n -th s -gonal number. Consider the Diophantine equation Ps(n)=tm for integers n,s,t and m>2. All solutions to this equation are known for m>2 and s∈{3,5,6,8,10,20}. Here we extend these results to the cases s=2k+4 (where k=4,6 or 5≤k≤97 is a prime number) and s=k+4 (where k=9,15 or 3≤k≤97 is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective abc conjecture) that there will be no additional solutions beyond those explicitly shown in Theorems 1, 2 and 3 .