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Displaying 141 – 160 of 314

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On the diophantine equation f(x)f(y) = f(z)²

Maciej Ulas (2007)

Colloquium Mathematicae

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 w+x+y = z, with wxyz = 2r 3s 5t.

L. J. Alex, L. L. Foster (1995)

Revista Matemática de la Universidad Complutense de Madrid

In this paper we complete the solution to the equation w+x+y = z, where w, x, y, and z are positive integers and wxyz has the form 2r 3s 5t, with r, s, and t non negative integers. Here we consider the case 1 < w ≤ x ≤ y, the remaining case having been dealt with in our paper: On the Diophantine equation 1+ X + Y = Z, Rocky Mountain J. of Math. This work extends earlier work of the authors in the field of exponential Diophantine equations.

On the Diophantine equation x² - dy⁴ = 1 with prime discriminant II

D. Poulakis, P. G. Walsh (2006)

Colloquium Mathematicae

Let p denote a prime number. P. Samuel recently solved the problem of determining all squares in the linear recurrence sequence {Tₙ}, where Tₙ and Uₙ satisfy Tₙ² - pUₙ² = 1. Samuel left open the problem of determining all squares in the sequence {Uₙ}. This problem was recently solved by the authors. In the present paper, we extend our previous joint work by completely solving the equation Uₙ = bx², where b is a fixed positive squarefree integer. This result also extends previous work of the second...

Currently displaying 141 – 160 of 314