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We consider $N$ , the number of solutions $(x, y, u, v)$ to the equation ${(- 1)}^{u} r a^{x} + {(- 1)}^{v} s b^{y} = c$ in nonnegative integers $x, y$ and integers $u, v \in {0, 1}$ , for given integers $a > 1$ , $b > 1$ , $c > 0$ , $r > 0$ and $s > 0$ . When $gcd (r a, s b) = 1$ , we show that $N \leq 3$ except for a finite number of cases all of which satisfy $max (a, b, r, s, x, y) < 2 \cdot 10^{15}$ for each solution; when $gcd (a, b) > 1$ , we show that $N \leq 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N = 3$ solutions.

The number of squarefull numbers in an interval

Hong-Quan Liu (1993)

Acta Arithmetica

The number of squares and $B_{h} [g]$ sets

Ben Green (2001)

Acta Arithmetica

The Number of Steps in a Finite Jacobi Algorithm.

Roland Fischer, Fritz Schweiger (1975)

Manuscripta mathematica

The number of subspaces occurring in the p-adic subspace theorem in diophantine approximation.

Hans Peter Schlickewei (1990)

Journal für die reine und angewandte Mathematik

The number of topologies on a finite set.

Benoumhani, Moussa (2006)

Journal of Integer Sequences [electronic only]

The number of unsieved integers up to x

Andrew Granville, K. Soundararajan (2004)

Acta Arithmetica

The number of zeroes of x3 + y3 + cz3 in certain finite fields.

S. Chowla, J. Cowles, M. Cowles (1978)

Journal für die reine und angewandte Mathematik

The number of zeros of polynomials in valuation rings of complete discretely valued fields

Andrzej Schinzel (1984)

Fundamenta Mathematicae

The number Γ(k) in Waring's problem

Christian Elsholtz (2008)

Acta Arithmetica

The numbers of sums of quadratic residues and of non-residues respectively taken n at a time and congruent to any given integer to an odd prime modulus p.

J.C. Fields (1893)

Journal für die reine und angewandte Mathematik

The Number-System of Algebra [Book]

Henry Burchard Fine (1891)

The number-theoretic content of the Jacobi triple product identity.

Wilf, Herbert S. (1999)

Séminaire Lotharingien de Combinatoire [electronic only]

The number-theoretic work of Paul Turán

G. Halász (1980)

Acta Arithmetica

The Oppenheim series expansions and Hausdorff dimensions

Jun Wu (2003)

Acta Arithmetica

The optimality of the Bounded Height Conjecture

Evelina Viada (2009)

Journal de Théorie des Nombres de Bordeaux

In this article we show that the Bounded Height Conjecture is optimal in the sense that, if $V$ is an irreducible subvariety with empty deprived set in a power of an elliptic curve, then every open subset of $V$ does not have bounded height. The Bounded Height Conjecture is known to hold. We also present some examples and remarks.

The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve

J. Cheon, S. Hahn (1999)

Acta Arithmetica

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