×

The ABC conjecture and the powerful part of terms in binary recurring sequences. (English) Zbl 0923.11033

Binary recurrence sequences have many interesting arithmetic properties like identities and divisibility properties. Under a certain assumption – the so-called ABC conjecture is true – such sequences with a positive discriminant have only finitely many terms which are powerful numbers. Here a natural number \(n\) is powerful if \(p^2\) divides \(n\) whenever a prime \(p\) divides \(n\).

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11D41 Higher degree equations; Fermat’s equation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Browkin, J.; Brzezinski, J., Some remarks on the abc conjecture, Math. Comp., 62, 931-939 (1994) · Zbl 0804.11006
[2] Granville, A., Some conjectures related to Fermat’s Last Theorem, (Mollin, R. A., Number Theory (1990), de Gruyter: de Gruyter Berlin/New York), 177-192 · Zbl 0702.11015
[3] Lang, S., Old and new conjectured diophantine inequalities, Bull. Amer. Math. Soc., 23, 37-75 (1990) · Zbl 0714.11034
[4] Lehmer, D. H., An extended theory of Lucas functions, Ann. of Math., 31, 419-448 (1930) · JFM 56.0874.04
[5] Masser, D. W., Open problems, (Chen, W. W.L., Proceedings Symp. Analytic Number Th. (1985), Imperial College: Imperial College London)
[6] Mollin, R. A., Masser’s conjecture used to prove results about powerful numbers, J. Math. Sci., 7, 29-32 (1996)
[7] Mollin, R. A., Quadratics (1996), CRC Press: CRC Press New York
[8] Nitaj, A., La conjecture abc, Enseign. Math., 42, 3-24 (1996) · Zbl 0856.11014
[9] Oesterlé, J., Nouvelles approches du “Théorème” de Fermat, Astérisque, 161-162, 165-186 (1988)
[10] Pethö, A., Perfect powers in second order linear recurrences, J. Number Theory, 15, 5-13 (1982) · Zbl 0488.10009
[12] Shorey, T. N.; Stewart, C. L., On the diophantine equation \(ax^{2t} + bx^ty cy^2\) d\), Math. Scand., 52, 24-36 (1983) · Zbl 0491.10016
[13] Shorey, T. N.; Tijdeman, R., Exponential Diophantine Equations (1986), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0606.10011
[14] Silverman, J. H., Wieferich’s condition and the abc conjecture, J. Number Theory, 30, 226-237 (1988) · Zbl 0654.10019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.