On the Skolem problem and some related questions for parametric families of linear recurrence sequences

Ostafe, Alina; Shparlinski, Igor E.

doi:10.4153/s0008414x21000080

Cited by 4 publications

(3 citation statements)

References 40 publications

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“…In Section 7, we will show how Theorem 3 gives a new method of determining if a linear recurrence sequence of order 3 contains zeroes. For further references of preexisting methods and results on this problem and related problems, we refer the reader to [14,27,32] and the references contained within them.…”

Section: 𝐺 = ∏mentioning

confidence: 99%

On the abc$abc$ conjecture in algebraic number fields

Scoones

2023

Mathematika

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In this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

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Section: 𝐺 = ∏mentioning

confidence: 99%

On the abc$abc$ conjecture in algebraic number fields

Scoones

2023

Mathematika

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“…It is not hard to derive this theorem from classical results on unlikely intersection like the Theorem of Bombieri-Masser-Zannier-Maurin [4,5,9]. See also the recent work of Ostafe and Shparlinski [10,11], especially Corollary 2.13 and Theorem 2.14 in [11].…”

Section: Introductionmentioning

confidence: 96%

Binary polynomial power sums vanishing at roots of unity

Bilu¹,

Luca²

2020

Preprint

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“…Bertók and Hajdu [3,4] proved that in some sense Skolem's conjecture is valid for 'almost all' equations. For strongly related problems and results concerning recurrence sequences, see the papers [10,11,8], and the references there.…”

Section: Introductionmentioning

confidence: 99%

Skolem’s conjecture confirmed for a family of exponential equations, II

2021

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According to Skolem's conjecture, if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriately chosen modulus. Besides several concrete equations, the conjecture has only been proved for rather special cases. In this paper we prove the conjecture for equations of the form x n − by k1 1 . . . y k ℓ ℓ = ±1, where b, x, y 1 , . . . , y ℓ are fixed integers and n, k 1 , . . . , k ℓ are non-negative integral unknowns. This result extends a recent theorem of Hajdu and Tijdeman.

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On the Skolem problem and some related questions for parametric families of linear recurrence sequences

Cited by 4 publications

References 40 publications

On the abc$abc$ conjecture in algebraic number fields

On the abc$abc$ conjecture in algebraic number fields

Binary polynomial power sums vanishing at roots of unity

Skolem’s conjecture confirmed for a family of exponential equations, II

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