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Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.

On the Diophantine equation F(x)=G(y)

Sz. Tengely (2003)

Acta Arithmetica

On the Diophantine equation f(y) - x = 0

Michael Fried (1971)

Acta Arithmetica

On the Diophantine equation $G_{n} (x) = G_{m} (P (x))$ for third order linear recurring sequences.

Fuchs, Clemens (2004)

Portugaliae Mathematica. Nova Série

On the Diophantine equation $\sum_{j = 1}^{k} j F_{j}^{p} = F_{n}^{q}$

Gökhan Soydan, László Németh, László Szalay (2018)

Archivum Mathematicum

Let $F_{n}$ denote the $n^{t h}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_{1}^{p} + 2 F_{2}^{p} + \dots + k F_{k}^{p} = F_{n}^{q}$ in the positive integers $k$ and $n$ , where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set ${1, 2}$ . Based on the specific cases we could solve, and a computer search with $p, q, k \leq 100$ we conjecture that beside the trivial solutions only $F_{8} = F_{1} + 2 F_{2} + 3 F_{3} + 4 F_{4}$ , $F_{4}^{2} = F_{1} + 2 F_{2} + 3 F_{3}$ , and $F_{4}^{3} = F_{1}^{3} + 2 F_{2}^{3} + 3 F_{3}^{3}$ satisfy the title equation.

On the Diophantine equation $k / n = a_{1} / x_{1} + a_{2} / x_{2} + a_{3} / x_{3}$

William A. Webb (1976)

Časopis pro pěstování matematiky

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

Peng Yang, Tianxin Cai (2012)

Acta Arithmetica

On the diophantine equation $(\binom{n}{k}) = x^{l}$

K. Győry (1997)

Acta Arithmetica

P. 294, line 14: For “Satz 8” read “Satz 7”, and for “equation (10)” read “equation (13)”.

On the diophantine equation $n (n + 1) . . . (n + k - 1) = b x^{l}$

K. Győry (1998)

Acta Arithmetica

On the Diophantine equation $\frac{q^{n} - 1}{q - 1} = y$

Amir Khosravi, Behrooz Khosravi (2003)

Commentationes Mathematicae Universitatis Carolinae

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.

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