An extension of Sylvester’s theorem on arithmetic progressions
Fecha
2023-06-18
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MDPI
Resumen
Sylvester’s theorem states that every number can be decomposed into a sum of consecutive
positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number
as a sum of consecutive integers. The first generalization we propose of the theorem characterizes
the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to
synthesizing and rediscovering known results, the method we propose allows us to state a second
generalization and characterize the partitions of a number into parts whose differences between
consecutive parts form an arithmetic progression. To achieve this, we will analyze the set of divisors
in arithmetics that modify the usual definition of the multiplication operation between two integers.
As we will see, symmetries arise in the set of divisors based on two parameters: t1, being even or odd,
and t2, congruent to 0, 1, or 2 (mod 3). This approach also leads to a unique representation result of
the same nature as Sylvester’s theorem, i.e., a power of 3 cannot be represented as a sum of three or
more terms of a positive integer sequence such that the differences between consecutive terms are
consecutive integers.
Descripción
Se generaliza un resultado de Sylvester en el que estudia la descomposición de un número como suma de consecutivos. En este artículo se estudia la descomposición de un número como suma de una serie que forma una progresión aritmética.
Citación
Munagi AO, de Vega FJ. An Extension of Sylvester’s Theorem on Arithmetic Progressions. Symmetry. 2023; 15(6):1276
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