Diophantine equations with Euler polynomials

Dijana Kreso; Csaba Rakaczki

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On some generalizations of the diophantine equation $s (1^{k} + 2^{k} + \dots + x^{k}) + r = d y^{n}$

Csaba Rakaczki (2012)

Acta Arithmetica

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On the Diophantine equation $X^{2} - (2^{2 m} + 1) Y^{4} = - 2^{2 m}$

Michael Stoll, P. G. Walsh, Pingzhi Yuan (2009)

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On the Diophantine equation $x^{p} + y^{2 p} = z^{2}$

A. Rotkiewicz, A. Schinzel (1987)

Colloquium Mathematicae

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A note on the Diophantine equation $x^{2} + q^{m} = y^{3}$

Hui Lin Zhu (2011)

Acta Arithmetica

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The diophantine equation $x^{2} + C = y^{n}$ , II

J. H. E. Cohn (2003)

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On the Diophantine equation $x^{2} + q^{2 m} = 2 y^{p}$

Sz. Tengely (2007)

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A note on the Diophantine equation $(a^{n} x^{m} \pm 1) / (a^{n} x \pm 1) = y^{n} + 1$

Jiagui Luo (2001)

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A note on the Diophantine equation P(z) = n! + m!

Maciej Gawron (2013)

Colloquium Mathematicae

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We consider the Brocard-Ramanujan type Diophantine equation P(z) = n! + m!, where P is a polynomial with rational coefficients. We show that the ABC Conjecture implies that this equation has only finitely many integer solutions when d ≥ 2 and $P (z) = a_{d} z^{d} + a_{d - 3} z^{d - 3} + \dots + a ₁ x + a ₀$ .

Searching for Diophantine quintuples

Mihai Cipu, Tim Trudgian (2016)

Acta Arithmetica

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We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most $5.441 \cdot 10^{26}$ Diophantine quintuples.

Solving diophantine equations $x^{4} + y^{4} = q z^{p}$

Luis V. Dieulefait (2005)

Acta Arithmetica

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On the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$

Florian Luca, Alain Togbé (2009)

Colloquium Mathematicae

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We find all the solutions of the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$ in positive integers x,y,α,β,n ≥ 3 with x and y coprime.

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

Min Tang, Quan-Hui Yang (2013)

Colloquium Mathematicae

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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.

On the Diophantine equation $(\binom{n}{k_{1}, . . ., k_{s}}) = x^{l}$

Peng Yang, Tianxin Cai (2012)

Acta Arithmetica

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On the Diophantine equation $x^{2} + 7 = y^{m}$

Samir Siksek, John E. Cremona (2003)

Acta Arithmetica

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Method of infinite ascent applied on $- (2^{p} \cdot A^{6}) + B^{3} = C^{2}$

Susil Kumar Jena (2013)

Communications in Mathematics

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In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for $(A, B, C)$ of the Diophantine equation $- (2^{p} \cdot A^{6}) + B^{3} = C^{2}$ for any positive integral values of $p$ when $p \equiv 1$ (mod 6) or $p \equiv 2$ (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.

A conjecture concerning the exponential diophantine equation $a^{x} + b^{y} = c^{z}$

Maohua Le (2003)

Acta Arithmetica

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