**s(n) An Arithmetic Function of Some Interest, and Related Arithmetic**

*Gideon Samid*

**Abstract: **Every integer n > 0 º N defines an increasing monotonic series of integers: n1, n2, ...nk, such that nk = nk +k(k-1)/2. We define as s(m) the number of such series that an integer m belongs to. We prove that there are infinite number of integers with s=1, all of the form 2^t (they belong only to the series that they generate, not to any series generated by a smaller integer). We designate them as s-prime integers. All integers with a factor other than 2 are not s-prime (s>1), but are s-composite. However, the arithmetic s function shows great variability, lack of apparent pattern, and it is conjectured that s(n) is unbound. Two integers, n and m, are defined as s-congruent if they have a common s-series. Every arithmetic equation can be seen as an expression without explicit unknowns -- provided every integer variable can be replaced by any s-congruent number. To validate the equation one must find a proper match of such members. This defines a special arithmetic, which appears well disposed towards certain cryptographic applications.

**Category / Keywords: **foundations / number theory

**Date: **received 3 Feb 2004

**Contact author: **samidg at tx technion ac il

**Available format(s): **PDF | BibTeX Citation

**Version: **20040216:094219 (All versions of this report)

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