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

Samir Siksek

Displaying similar documents to “On the diophantine equation $x^{2} = y^{p} + 2^{k} z^{p}$ ”

Diophantine $m$ -tuples and elliptic curves

Andrej Dujella (2001)

Journal de théorie des nombres de Bordeaux

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A Diophantine $m$ -tuple is a set of $m$ positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form $y^{2} = (a x + 1) (b x + 1) (c x + 1)$ , where ${a, b, c}$ , is a Diophantine triple. In particular, we consider the elliptic curve $E_{k}$ defined by the equation $y^{2} = (F_{2 k} x + 1) (F_{2 k + 2} x + 1) (F_{2 k + 4} x + 1),$ where $k \geq 2$ and $F_{n}$ , denotes the $n$ -th Fibonacci number. We prove that if the rank of $E_{k} (𝐐)$ is equal to one, or $k \leq 50$ , then all integer points on $E_{k}$ are given by $\begin{matrix} (x, y) \in {(0 \pm 1), (4 F_{2 k + 1} F_{2 k + 2} F_{2 k + 3} \pm (2 F_{2 k + 1} F_{2 k + 2} - 1) \\ \times (2 F_{2 k + 2}^{2} + 1) (2 F_{2 k + 2} F_{2 k + 3} + 1)} . \end{matrix}$

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

Samir Siksek, John E. Cremona (2003)

Acta Arithmetica

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Cubic forms, powers of primes and the Kraus method

Andrzej Dąbrowski, Tomasz Jędrzejak, Karolina Krawciów (2012)

Colloquium Mathematicae

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We consider the Diophantine equation $(x + y) (x ² + B x y + y ²) = D z^{p}$ , where B, D are integers (B ≠ ±2, D ≠ 0) and p is a prime >5. We give Kraus type criteria of nonsolvability for this equation (explicitly, for many B and D) in terms of Galois representations and modular forms. We apply these criteria to numerous equations (with B = 0, 1, 3, 4, 5, 6, specific D’s, and p ∈ (10,10⁶)). In the last section we discuss reductions of the above Diophantine equations to those of signature (p,p,2).

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 $\frac{q^{n} - 1}{q - 1} = y$

Amir Khosravi, Behrooz Khosravi (2003)

Commentationes Mathematicae Universitatis Carolinae

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There exist many results about the Diophantine equation $(q^{n} - 1) / (q - 1) = y^{m}$ , where $m \geq 2$ and $n \geq 3$ . In this paper, we suppose that $m = 1$ , $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y - 1$ is less than or equal to $3$ . Then we solve completely the Diophantine equation $(q^{n} - 1) / (q - 1) = y$ for infinitely many values of $y$ . This result finds frequent applications in the theory of finite groups.

A ternary Diophantine inequality over primes

Roger Baker, Andreas Weingartner (2014)

Acta Arithmetica

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Let 1 < c < 10/9. For large real numbers R > 0, and a small constant η > 0, the inequality $| p ₁^{c} + p ₂^{c} + p ₃^{c} - R | < R^{- η}$ holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].

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

Luis V. Dieulefait (2005)

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.

The method of infinite ascent applied on $A^{4} \pm n B^{3} = C^{2}$

Susil Kumar Jena (2013)

Czechoslovak Mathematical Journal

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Each of the Diophantine equations $A^{4} \pm n B^{3} = C^{2}$ has an infinite number of integral solutions $(A, B, C)$ for any positive integer $n$ . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when $A$ , $B$ and $C$ are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions $(A, B, C)$ of the Diophantine equation $a A^{3} + c B^{3} = C^{2}$ for any co-prime integer pair $(a, c)$ . ...

On a ternary Diophantine problem with mixed powers of primes

Alessandro Languasco, Alessandro Zaccagnini (2013)

Acta Arithmetica

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Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality $| λ ₁ p ₁ + λ ₂ p ² ₂ + λ ₃ p ₃^{k} + ϖ | \leq {(m a x_{j} p_{j})}^{- (33 - 29 k) / (72 k) + ε}$ has infinitely many solutions in prime variables p₁, p₂, p₃.

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

J. H. E. Cohn (2003)

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

Hui Lin Zhu (2011)

Acta Arithmetica

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Displaying similar documents to “On the diophantine equation x 2 = y p + 2 k z p ”

Displaying similar documents to “On the diophantine equation $x^{2} = y^{p} + 2^{k} z^{p}$ ”