On squares of squares II
On the computation of the class numbers of some cubic fields.
On the congruence f(x) + g(y) + c ≡ 0 (mod xy) (completion of Mordell's proof)
Assertions on the congruence f(x) + g(y) + c ≡ 0 (mod xy) made without proof by Mordell in his paper in Acta Math. 88 (1952) are either proved or disproved.
On the difference of cubes (mod p)
On the Diophantine equation 1 + x + y = z, with xyz = 2r3s7t.
On the diophantine equation 27 y2 + 4 x3 = D.
On the diophantine equation 27y2 + 4x3 = M.
On the Diophantine Equation 2x3 + 1 = py2.
On the Diophantine Equation Ax4 -By2 = C (C = 1, 4).
On the diophantine equation f(x)f(y) = f(z)²
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)
On the diophantine equation w+x+y = z, with wxyz = 2r 3s 5t.
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
On the diophantine equation
On the Diophantine equation .
On the diophantine equation .
On the Diophantine equation (x² ± C)(y² ± D) = z⁴
On the Diophantine equation x² - dy⁴ = 1 with prime discriminant II
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...
On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement.
On the equation . (Sur l'équation .)