On the set of solutions of the system
A proof is given that the system in the title has infinitely many solutions of the form , where and are rational numbers.
On and
The two related Diophantine equations: and , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.
On
In p. 219 of R.K. Guy’s Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation has no integer solutions with and . But, contrary to this expectation, we show that for , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition .
On zeros of diagonal forms over p-adic fields
On zeros of forms over local fields
Pairs of additive equations IV. Sextic equations
Pairs of additive equations of small degree
Pairs of additive forms and Artin's conjecture
Pairs of cubic forms in many variables
Parametric Solutions of the Diophantine Equation A² + nB⁴ = C³
The Diophantine equation A² + nB⁴ = C³ has infinitely many integral solutions A, B, C for any fixed integer n. The case n = 0 is trivial. By using a new polynomial identity we generate these solutions, and then give conditions when the solutions are pairwise co-prime.
Points de hauteur bornée sur les hypersurfaces lisses des variétés toriques
We demonstrate the Batyrev-Manin Conjecture for the number of points of bounded height on hypersurfaces of some toric varieties whose rank of the Picard group is 2. The method used is inspired by the one developed by Schindler for the case of hypersurfaces of biprojective spaces and by Blomer and Brüdern for some hypersurfaces of multiprojective spaces. These methods are based on the Hardy-Littlewood circle method. The constant obtained in the final asymptotic formula is the one conjectured by Peyre....
Problème de Waring avec exposant non entier
Problème de Waring avec exposants non entiers
Problème de Waring pour les bicarrés : le point en 1984
Products of consecutive integers and the Markoff equation.
Rational points on certain quintic hypersurfaces
Rational points on some pencils of conics with 6 singular fibres
Simultaneous diagonal equations over 𝔭-adic fields
Simultaneous p-adic Zeros of Quadratic Forms.
Simultaneous quadratic equations II