Paper 2004/034

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.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
Published elsewhere. Unknown where it was published
Keywords
number theory
Contact author(s)
samidg @ tx technion ac il
History
2004-02-16: received
Short URL
https://ia.cr/2004/034
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2004/034,
      author = {Gideon Samid},
      title = {s(n) An Arithmetic Function of Some Interest, and Related Arithmetic},
      howpublished = {Cryptology {ePrint} Archive, Paper 2004/034},
      year = {2004},
      url = {https://eprint.iacr.org/2004/034}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.