On the Diophantine Equation Cx2 + D = 2yn.
On the diophantine equation f (x, y) = 0.
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 f(y) - x = 0
On the Diophantine equation
On the diophantine equation
On the diophantine equation
Consider the equation in the title in unknown integers with , , , , and . Under the above conditions we give all solutions of the title equation (see Theorem 1).
On the diophantine equation
On the Diophantine equation
On the diophantine equation
Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either , k ≥ 1 or if c ≥ 2.
On the Diophantine Equation x2 + D = 4yq.
On the diophantine equation xl + yl = czln.
On the diophantine equation x(x-1)...(x-(m-1)) = λy(y-1 )...(y-(n-1)) + l
On the diophantine equation
On the diophantine equations 2y2 = 7k + 1 and x2 + 11 = 3n.
On the equation
On the equation . (Sur l'équation .)
On the equation
We discuss the equation in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and α is a positive integer. The technique used to prove Fermat’s Last Theorem shows that the equation has no solutions with α < 1 or b even. When α=1 and b is odd, there are the two trivial solutions (±1, ∓ 1, ±1). In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for p≡ 1 mod 4.
On the equation