Powerful arithmetic progressions
A \textit{powerful arithmetic progression} is a progression whose \(k\)th term is a perfect \(k\)th power for all \(k\). An arithmetic progression \(a_1\), \(a_2\), \dots\ is called \textit{primitive} if \(\gcd(a_1,a_2)=1\). In 1998, K. D. Bolkan asked the following question: what is the length of the longest nonconstant powerful arithmetic progression? In 2000, J. P. Robertson proved that the answer is five. In the present paper, the author proves in particular the following results: -- The only primitive powerful arithmetic progression of length five is the trivial one 1,~1,~1,~1,~1. -- The are infinitely many powerful arithmetic progressions of length four. -- The are infinitely many pairwise nonproportional powerful arithmetic progressions of length five. The proof of the first of these results is obtained by a deep study of the Diophantine equation \(3X^4-Y^2=2Z^5\), involving in particular the elliptic Chabauty method.
scientific article; zbMATH DE number 5619293
scientific article