The Generalized Nagell–Ljunggren Problem: Powers with Repetitive Representations. Issue 4 (2nd October 2019)

• Record Type: Journal Article
• Authors: Bridy, Andrew; Oliver, Robert J. Lemke; Shallit, Arlo; Shallit, Jeffrey
• Is Part Of: Experimental mathematics. Volume 28:Issue 4(2019)
• Page Start: 428
• Page End: 439
• Publication Date: 2019-10-02
• Subjects: Nagell–Ljunggren problem -- Diophantine equation -- base-b representation -- repetition -- power -- abc-conjecture; Primary 11D61 -- Secondary 11A63 -- 11Y50 -- 11D41; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2017.1419391
• Languages: English
• ISSNs: 1058-6458

Abstract

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.