Solutions rationelles de certaines équations fonctionelles.
Solutions rationnelles d'équations différentielles. Application à un problème de transcendance
Solutions to conjectures on a nonlinear recursive equation
We obtain solutions to some conjectures about the nonlinear difference equation More precisely, we get not only a condition under which the equilibrium point of the above equation is globally asymptotically stable but also a condition under which the above equation has a unique positive cycle of prime period two. We also prove some further results.
Solutions to infinite families of complex cubic Thue equations.
Solutions to xyz = 1 and x + y + z = k in algebraic integers of small degree, II
Let k ∈ ℤ be such that is finite, where . We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.
Solutions to xyz = 1 and x+y+z = k in algebraic integers of small degree, I
Let k ∈ ℤ be such that , where . We determine all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.
Solvability of a Diophantine inequality in algebraic number fields
Solvability of 𝔭-adic diagonal equations
Solving a ± b = 2c in elements of finite sets
We show that if A and B are finite sets of real numbers, then the number of triples (a,b,c) ∈ A × B × (A ∪ B) with a + b = 2c is at most (0.15+o(1))(|A|+|B|)² as |A| + |B| → ∞. As a corollary, if A is antisymmetric (that is, A ∩ (-A) = ∅), then there are at most (0.3+o(1))|A|² triples (a,b,c) with a,b,c ∈ A and a - b = 2c. In the general case where A is not necessarily antisymmetric, we show that the number of triples (a,b,c) with a,b,c ∈ A and a - b = 2c is at most (0.5+o(1))|A|². These estimates...
Solving a family of Thue equations with an application to the equation x²-Dy⁴=1
Solving a linear equation in a set of integers I
Solving a linear equation in a set of integers II
Solving an exponential Diophantine equation
Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas
This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.
Solving conics over function fields
Let be a field whose characteristic is not and . We give a simple algorithm to find, given , a nontrivial solution in (if it exists) to the equation . The algorithm requires, in certain cases, the solution of a similar equation with coefficients in ; hence we obtain a recursive algorithm for solving diagonal conics over (using existing algorithms for such equations over ) and over .
Solving diophantine equations
Solving elliptic diophantine equations avoiding Thue equations and elliptic logarithms.
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms
Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations
Solving elliptic diophantine equations: the general cubic case