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Displaying 21 – 40 of 148

The Diophantine equation $A X^{2} - B Y^{2} = C$ solved via continued fractions.

Mollin, R.A., Cheng, K., Goddard, B. (2002)

Acta Mathematica Universitatis Comenianae. New Series

The Diophantine equation ${(b n)}^{x} + {(2 n)}^{y} = {((b + 2) n)}^{z}$

Min Tang, Quan-Hui Yang (2013)

Colloquium Mathematicae

Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation $b^{x} + 2^{y} = {(b + 2)}^{z}$ has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

The Diophantine equation $D x ² + 2^{2 m + 1} = y ⁿ$

J. H. E. Cohn (2003)

Colloquium Mathematicae

It is shown that for a given squarefree positive integer D, the equation of the title has no solutions in integers x > 0, m > 0, n ≥ 3 and y odd, nor unless D ≡ 14 (mod 16) in integers x > 0, m = 0, n ≥ 3, y > 0, provided in each case that n does not divide the class number of the imaginary quadratic field containing √(-2D), except for a small number of (stated) exceptions.

The diophantine equation $d y^{2} = a x^{4} + b x^{2} + c$

L. Mordell (1969)

Acta Arithmetica

The Diophantine equation f(x) = g(y)

Yuri Bilu, Robert Tichy (2000)

Acta Arithmetica

The diophantine equation $n^{i} + 1 = k (d n - 1)$ .

Ligh, Steve, Bourque, Keith (1989)

International Journal of Mathematics and Mathematical Sciences

The diophantine equation $r^{2} + r (x + y) = k x y$ .

Utz, W.R. (1985)

International Journal of Mathematics and Mathematical Sciences

The diophantine equation $x^{2} + 2^{a} \cdot 17^{b} = y^{n}$

Su Gou, Tingting Wang (2012)

Czechoslovak Mathematical Journal

Let $ℤ$ , $ℕ$ be the sets of all integers and positive integers, respectively. Let $p$ be a fixed odd prime. Recently, there have been many papers concerned with solutions $(x, y, n, a, b)$ of the equation $x^{2} + 2^{a} p^{b} = y^{n}$ , $x, y, n \in ℕ$ , $gcd (x, y) = 1$ , $n \geq 3$ , $a, b \in ℤ$ , $a \geq 0$ , $b \geq 0 .$ And all solutions of it have been determined for the cases $p = 3$ , $p = 5$ , $p = 11$ and $p = 13$ . In this paper, we mainly concentrate on the case $p = 3$ , and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions $(x, y, n, a, b)$ of the equation $x^{2} + 2^{a} \cdot 17^{b} = y^{n}$ , $x, y, n \in ℕ$ , $gcd (x, y) = 1$ , $n \geq 3$ , $a, b \in ℤ$ , $a \geq 0$ , $b \geq 0$ , are determined....