Diophantine $m$-tuples and elliptic curves

Andrej Dujella

Displaying similar documents to “Diophantine $m$ -tuples and elliptic curves”

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

Samir Siksek, John E. Cremona (2003)

Acta Arithmetica

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The integral points on elliptic curves $y^{2} = x^{3} + (36 n^{2} - 9) x - 2 (36 n^{2} - 5)$

Hai Yang, Ruiqin Fu (2013)

Czechoslovak Mathematical Journal

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Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n > 1$ and both $6 n^{2} - 1$ and $12 n^{2} + 1$ are odd primes, then the general elliptic curve $y^{2} = x^{3} + (36 n^{2} - 9) x - 2 (36 n^{2} - 5)$ has only the integral point $(x, y) = (2, 0)$ . By this result we can get that the above elliptic curve has only the trivial integral point for $n = 3, 13, 17$ etc. Thus it can be seen that the elliptic curve $y^{2} = x^{3} + 27 x - 62$ really is an unusual elliptic curve which has large integral points. ...

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

Samir Siksek (2003)

Journal de théorie des nombres de Bordeaux

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We attack the equation of the title using a Frey curve, Ribet’s level-lowering theorem and a method due to Darmon and Merel. We are able to determine all the solutions in pairwise coprime integers $x, y, z$ if $p \geq 7$ is prime and $k \geq 2$ . From this we deduce some results about special cases of this equation that have been studied in the literature. In particular, we are able to combine our result with previous results of Arif and Abu Muriefah, and those of Cohn to obtain a complete solution for the equation...

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 $P (z) = a_{d} z^{d} + a_{d - 3} z^{d - 3} + \dots + a ₁ x + a ₀$ .

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

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

Acta Arithmetica

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

Peng Yang, Tianxin Cai (2012)

Acta Arithmetica

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

J. H. E. Cohn (2003)

Acta Arithmetica

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

Jiagui Luo (2001)

Acta Arithmetica

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

Sz. Tengely (2007)

Acta Arithmetica

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

J. H. E. Cohn (2003)

Colloquium Mathematicae

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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.

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.

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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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.

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.

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

Maohua Le (2003)

Acta Arithmetica

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On the diophantine equation $a x ² + b^{m} = 4 y ⁿ$

S. Akhtar Arif, Amal S. Al-Ali (2002)

Acta Arithmetica

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

Luis V. Dieulefait (2005)

Acta Arithmetica

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Further remarks on Diophantine quintuples

Mihai Cipu (2015)

Acta Arithmetica

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A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < e $s a t i s f i e s$ d < 1.55·1072 $a n d$ b < 6.21·1035 $w h e n 4 a < b, w h i l e f o r b < 4 a o n e h a s e i t h e r$ c = a + b + 2√(ab+1)...

Displaying similar documents to “Diophantine m -tuples and elliptic curves”

Displaying similar documents to “Diophantine $m$ -tuples and elliptic curves”