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Ternary Diophantine Equations via Galois Representations and Modular Forms
In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2 , based upon the theory of Galois representations and modular forms. We subse- quentlyExpand
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Applications of the Hypergeometric Method to the Generalized Ramanujan-Nagell Equation
AbstractIn this paper, we refine work of Beukers, applying results from the theory of Padé approximation to (1 − z)1/2 to the problem of restricted rational approximation to quadratic irrationals. AsExpand
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Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1
Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional ``hypergeometric method'' for rational and algebraic approximationExpand
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Powers from Products of Consecutive Terms in Arithmetic Progression
We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power.Expand
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On the Diophantine equation
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Ternary Diophantine equations of signature (p, p, 3)
In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn + Byn = Cz3 for various choices of coefficients (A,B, C). As a byproduct of this, we show, if p is prime,Expand
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Fractional parts of powers of rational numbers
The author uses Pade approximation techniques and an elementary lemma on primes dividing binomial coefficients to sharpen a theorem of F. Beukers on fractional parts of powers of rationals. InExpand
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On the Diophantine Equation 1
In this paper, we resolve a conjecture of Schäffer on the solvability of Diophantine equations of the shape 1 k + 2 k + · · · + x k = y n , for 1 ≤ k ≤ 11. Our method, which may, with a modicum ofExpand
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Lucas' square pyramid problem revisited
are given by (s, t) = (1, 1) and (24, 70). Putative solutions by Moret-Blanc [30] and Lucas [25] contain fatal flaws (see e.g. [39] for details) and it was not until 1918 that Watson [39] was able toExpand
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The representation of integers by binary additive forms
Let a, b and n be integers with ≥ 3. We show that, in the sense of natural density, almost all integers represented by the binary form axn − byn are thus represented essentially uniquely. ByExpand
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