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

Su Gou; Tingting Wang

Displaying similar documents to “The diophantine equation $x^{2} + 2^{a} \cdot 17^{b} = y^{n}$ ”

Congruent numbers with higher exponents

Florian Luca, László Szalay (2006)

Acta Mathematica Universitatis Ostraviensis

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This paper investigates the system of equations $x^{2} + a y^{m} = z_{1}^{2}, x^{2} - a y^{m} = z_{2}^{2}$ in positive integers $x$ , $y$ , $z_{1}$ , $z_{2}$ , where $a$ and $m$ are positive integers with $m \geq 3$ . In case of $m = 2$ we would obtain the classical problem of congruent numbers. We provide a procedure to solve the simultaneous equations above for a class of the coefficient $a$ with the condition $gcd (x, z_{1}) = gcd (x, z_{2}) = gcd (z_{1}, z_{2}) = 1$ . Further, under same condition, we even prove a finiteness theorem for arbitrary nonzero $a$ .

Jeśmanowicz' conjecture with congruence relations

Yasutsugu Fujita, Takafumi Miyazaki (2012)

Colloquium Mathematicae

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Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b \equiv 0 (m o d 2^{r})$ and $b \equiv \pm 2^{r} (m o d a)$ for some non-negative integer r, then the Diophantine equation $a^{x} + b^{y} = c^{z}$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).

Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

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Given a binary recurrence ${u_{n}}_{n \geq 0}$ , we consider the Diophantine equation $u_{n_{1}}^{x_{1}} \dots u_{n_{L}}^{x_{L}} = 1$ with nonnegative integer unknowns $n_{1}, . . ., n_{L}$ , where $n_{i} \neq n_{j}$ for 1 ≤ i < j ≤ L, $m a x | x_{i} | : 1 \leq i \leq L \leq K$ , and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

On the Lebesgue-Nagell equation

Andrzej Dąbrowski (2011)

Colloquium Mathematicae

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We completely solve the Diophantine equations $x ² + 2^{a} q^{b} = y ⁿ$ (for q = 17, 29, 41). We also determine all $C = p ₁^{a ₁} \dots p_{k}^{a_{k}}$ and $C = 2^{a ₀} p ₁^{a ₁} \dots p_{k}^{a_{k}}$ , where $p ₁, . . ., p_{k}$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).

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)...

The diophantine equation $x^{2} = p^{a} \pm p^{b} + 1$

Florian Luca (2004)

Acta Arithmetica

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Common terms in binary recurrences

Erzsébet Orosz (2006)

Acta Mathematica Universitatis Ostraviensis

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The purpose of this paper is to prove that the common terms of linear recurrences $M (2 a, - 1, 0, b)$ and $N (2 c, - 1, 0, d)$ have at most $2$ common terms if $p = 2$ , and have at most three common terms if $p > 2$ where $D$ and $p$ are fixed positive integers and $p$ is a prime, such that neither $D$ nor $D + p$ is perfect square, further $a, b, c, d$ are nonzero integers satisfying the equations $a^{2} - D b^{2} = 1$ and $c^{2} - (D + p) d^{2} = 1$ .

On $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$

Susil Kumar Jena (2015)

Communications in Mathematics

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The two related Diophantine equations: $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$ , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.

Displaying similar documents to “The diophantine equation x 2 + 2 a · 17 b = y n ”

Displaying similar documents to “The diophantine equation $x^{2} + 2^{a} \cdot 17^{b} = y^{n}$ ”