An Extension of Sylvester’s Theorem on Arithmetic Progressions

Abstract

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.