The2-adic and3-adic valuation of the Tripell sequence and an application

Bravo, Jhon J.; Díaz, Maribel; Ramírez, José L.

doi:10.3906/mat-1907-40

Cited by 4 publications

(3 citation statements)

References 6 publications

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“…Basically, it shows how to use certain kinds of polynomial congruences to prove identities for linear recurrence sequences. Applying this technique, in [2], we obtained the following identity involving Tripell numbers. This result plays a crucial role in the proof of Theorem 2.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

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On a variant of the Brocard--Ramanujan equation and an application

Bravo¹,

Díaz²,

Ramírez³

2021

Publ. Math. Debrecen

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In this paper, we study the variant of the Brocard-Ramanujan diophantine equation m! + 1 = u 2 , where u is a member of a sequence of positive integers. Under some technical conditions on the sequence, we prove that this equation has at most finitely many solutions in positive integers m and u. As an application, we completely solve this equation when u is a Tripell number. The Tripell numbers are defined by the recurrence relation Tn = 2Tn−1 + Tn−2 + Tn−3 for n ≥ 3, with T0 = 0, T1 = 1 and T2 = 2 as initial conditions. This is known as the Brocard-Ramanujan diophantine equation, and it is still an open problem (see [11]). It is expected that the only solutions are (m, n) = (4, 5), (5, 11), (7, 71). Computations by Berndt and Galway [1] showed that there are no other solution in the range n < 10 9 . In 1993, Overholt [20] proved that the weak form of Szpiro's conjecture implies that equation (1) has

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Section: Auxiliary Resultsmentioning

confidence: 99%

“…He proved that if A is not a square, then equation (2) has only finitely many integral solutions. He also proved that if A is a square, then the weak form of Szpiro's conjecture implies the finiteness of solutions for equation (2). On the other hand, Luca [17] proved that the ABCconjecture implies that the diophantine equation…”

Section: Introductionmentioning

confidence: 99%

On a variant of the Brocard--Ramanujan equation and an application

Bravo¹,

Díaz²,

Ramírez³

2021

Publ. Math. Debrecen

View full text Add to dashboard Cite

show abstract

“…The more complicated case of k ≥ 5 odd and the shifted values T n (k) + 1 were studied by Young [39,40]. Results by Bravo, Díaz, and Ramírez [6] on so-called Tripell sequence imply that for p = 2, 3 the corresponding sequences of p-adic valuations are p-regular.…”

Section: Introductionmentioning

confidence: 99%

On $k$-regularity of sequences of valuations and last nonzero digits

Sobolewski¹

2022

Preprint

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Let b ≥ 2 be an integer base with prime factors p1, . . . , ps. In this paper we study sequences of "b-adic valuations" and last nonzero digits in b-adic expansions of the values f (n) = (f1(n), . . . , fs(n)), where each fi is a pi-adic analytic function. We give a complete classification concerning k-regularity of these sequences, which generalizes a result for b prime obtained by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on b-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely which terms of these sequences can be represented by certain ternary quadratic forms.

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On k-regularity of sequences of valuations and last non-zero digits

Sobolewski

2023

Ramanujan J

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Let $$b \ge 2$$ b ≥ 2 be an integer base with prime factors $$p_1, \ldots , p_s$$ p 1 , … , p s . In this paper we study sequences of “b-adic valuations” and last nonzero digits in b-adic expansions of the values $$f(n) = (f_1(n), \ldots , f_s(n))$$ f ( n ) = ( f 1 ( n ) , … , f s ( n ) ) , where each $$f_i$$ f i is a $$p_i$$ p i -adic analytic function. We give a complete classification concerning k-regularity of these sequences, which generalizes a result obtained for b prime by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on b-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely which terms of these sequences can be represented as a sum of three squares.

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The2-adic and3-adic valuation of the Tripell sequence and an application

Cited by 4 publications

References 6 publications

On a variant of the Brocard--Ramanujan equation and an application

On a variant of the Brocard--Ramanujan equation and an application

On $k$-regularity of sequences of valuations and last nonzero digits

On k-regularity of sequences of valuations and last non-zero digits

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