Exponential Diophantine Equations

Shorey, T. N.; Tijdeman, R.

doi:10.1017/cbo9780511566042

Cited by 338 publications

(269 citation statements)

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“…, i?t are multiplicatively independent, and s -1 = r + <, so their corresponding vectors are linearly independent. Then, by Lemma The usual regulator argument (for example [6], p.103), already alluded to, does not yield an upper bound for the fc ; -because the elements ni, . .…”

Section: V€s Vesmentioning

confidence: 97%

On the generators of S-unit groups in algebraic number fields

Brindza

1991

Bull. Austral. Math. Soc.

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Section: V€s Vesmentioning

confidence: 97%

On the generators of S-unit groups in algebraic number fields

Brindza

1991

Bull. Austral. Math. Soc.

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“…For various generalizations and related results, we refer to [20] and [24]. In the case when / is monic and its discriminant D(f) is different from zero, it follows from a theorem of Gy6ry [9] (see also Lemma 3 in Section 4) that In our paper, we shall deal with equation (13) in the important special case when / is irreducible.…”

Section: Super-elliptic Equationsmentioning

confidence: 99%

Bounds for the solutions of some diophantine equations in terms of discriminants

1991

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Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

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“…The proof of Theorem 2 uses the theory of lower bounds for linear forms in logarithms of algebraic numbers, and a very good introduction to this topic is [11]. Since such lower bounds usually involve some astronomical constants, the constant c 1 turns out to be very large, while the constant c 2 turns out to be very small.…”

Section: Cullen Numbers Asmentioning

confidence: 99%

Cullen Numbers in Binary Recurrent Sequences

Luca

Stănică

2004

Applications of Fibonacci Numbers

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In this paper, we prove that the generalized Cullen numbers, C n (s, l) := s n · 2 n + l, where l is an integer and s := (s n ) n≥0 is a sequence of integers satisfying log |s n | < 2 n−1 + O(1), occur only finitely many times in binary recurrent sequences (u n ) n≥0 whose characteristic roots are quadratic units and that satisfy some additional conditions. We also generalize this result in some sense to show that if we take any finite set of prime numbers P and any integer l, and we write u n − l = P Q, where P is a product of powers of the primes from P, and Q is free of primes from P, then there exist two computable constants c 1 and c 2 depending only on the sequence (u n ) n≥0 , the number l, and the given set of primes P, such that for n > c 1 we have log |Q| > |P | c2 .Finally, we find all Cullen numbers, i.e., numbers of the form n · 2 n + 1 and all Woodall numbers, i.e., the numbers of the form n · 2 n − 1, that are either Fibonacci or Pell

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Exponential Diophantine Equations

Cited by 338 publications

References 0 publications

On the generators of S-unit groups in algebraic number fields

On the generators of S-unit groups in algebraic number fields

Bounds for the solutions of some diophantine equations in terms of discriminants

Cullen Numbers in Binary Recurrent Sequences

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