The Generalized Nagell–Ljunggren Problem: Powers with Repetitive Representations. Issue 4 (2nd October 2019) - Bridy, Andrew; Oliver, Robert J. Lemke; Shallit, Arlo; Shallit, Jeffrey
ABSTRACT: We consider a natural generalization of the Nagell–Ljunggren equation to the case where the q th power of an integer y, for q ⩾ 2, has a base- b representation that consists of a length-ℓ block of digits repeated n times, where n ⩾ 2. Assuming the abc conjecture of Masser and Oesterlé, we completely characterize those triples ( q, n, ℓ) for which there are infinitely many solutions b . In all cases predicted by the abc conjecture, we are able (without any assumptions) to prove there are indeed infinitely many solutions.