An Application of van der Waerden's Theorem in Additive Number Theory

A sequence on a finite set of symbols is called strongly non-repetitive if no two adjacent (finite) segments are permutations of each other. Replacing the finite set of symbols of a strongly non-repetitive sequence by different prime numbers, one gets an infinite sequence on a finite set of integers such that no two adjacent segments have the same product. It is known that there are infinite strongly non-repetitive sequences on just four symbols. The aim of this paper is to show that there is no infinite sequence on a finite set of integers such that no two adjacent segments have the same sum. Thus, in the statement above, one cannot replace “product” by “sum”. Further we suggest some strengthened versions of the notion of strongly non-repetitive.

Related Organizations

University of Alabama at Birmingham
United States
University of California, Berkeley
United States

Keywords

van der Waerden's theorem, Special sequences and polynomials

Impact byBIP!

	selected citations These citations are derived from selected sources. This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).	1
	popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.	Average
	influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).	Average
	impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.	Average