A note on the diophantine equation $x^2+b^Y=c^z$

Maohua Le

Displaying similar documents to “A note on the diophantine equation $x^{2} + b^{Y} = c^{z}$ ”

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).

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 $a$ .

Note on the two congruences $a x^{2} + b y^{2} + e \equiv 0$ , $a x^{2} + b y^{2} + c z^{2} + d w^{2} \equiv 0 (mod. p)$ , where $p$ is an odd prime and $a \neg \equiv 0$ , $b \neg \equiv 0$ , $c \neg \equiv 0$ , $d \neg \equiv 0 (mod. p)$

Haridas Bagchi (1949)

Rendiconti del Seminario Matematico della Università di Padova

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Congruences for $q^{[p / 8]} (m o d p)$

Zhi-Hong Sun (2013)

Acta Arithmetica

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Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p / 8]} (m o d p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.

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

Su Gou, Tingting Wang (2012)

Czechoslovak Mathematical Journal

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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 $x^{2} + 2^{a} \cdot 17^{b} = y^{n}$ , $x, y, n \in ℕ$ , $gcd (x, y) = 1$ , $n \geq 3$ , $a, b \in ℤ$ , $a \geq 0$ ,...

Displaying similar documents to “A note on the diophantine equation x 2 + b Y = c z ”

Jeśmanowicz' conjecture with congruence relations

Congruent numbers with higher exponents

Note on the two congruences a x 2 + b y 2 + e ≡ 0 , a x 2 + b y 2 + c z 2 + d w 2 ≡ 0 ( mod. p ) , where p is an odd prime and a ¬ ≡ 0 , b ¬ ≡ 0 , c ¬ ≡ 0 , d ¬ ≡ 0 ( mod. p )

Congruences for q [ p / 8 ] ( m o d p )

The diophantine equation x 2 + 2 a · 17 b = y n

Displaying similar documents to “A note on the diophantine equation $x^{2} + b^{Y} = c^{z}$ ”

Note on the two congruences $a x^{2} + b y^{2} + e \equiv 0$ , $a x^{2} + b y^{2} + c z^{2} + d w^{2} \equiv 0 (mod. p)$ , where $p$ is an odd prime and $a \neg \equiv 0$ , $b \neg \equiv 0$ , $c \neg \equiv 0$ , $d \neg \equiv 0 (mod. p)$

Congruences for $q^{[p / 8]} (m o d p)$

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