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
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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
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