Artículo
Weak Schur numbers and the search for G.W. Walker’s lost partitions
Autor/es | Eliahou, Shalom
Marín Sánchez, Juan Manuel Revuelta Marchena, María Pastora ![]() ![]() ![]() ![]() ![]() ![]() ![]() Sanz Domínguez, María Isabel |
Departamento | Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII) |
Fecha de publicación | 2012 |
Fecha de depósito | 2022-09-01 |
Publicado en |
|
Resumen | A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such
that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n}
admits a partition into k weakly ... A set A of integers is weakly sum-free if it contains no three distinct elements x, y, z such that x + y = z. Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without proof. Here we show WS(5) ≥ 196, by constructing a partition of {1, . . . , 196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k = 6, we obtain WS(6) ≥ 572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver. |
Cita | Eliahou, S., Marín Sánchez, J.M., Revuelta Marchena, M.P. y Sanz Domínguez, M.I. (2012). Weak Schur numbers and the search for G.W. Walker’s lost partitions. Computers and Mathematics with Applications, 63 (1), 175-182. |
Ficheros | Tamaño | Formato | Ver | Descripción |
---|---|---|---|---|
1-s2.0-S0898122111009722-main.pdf | 308.7Kb | ![]() | Ver/ | |