Paper 2004/034

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

Gideon Samid


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.

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Published elsewhere. Unknown where it was published
number theory
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samidg @ tx technion ac il
2004-02-16: received
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      author = {Gideon Samid},
      title = {s(n) An Arithmetic Function of Some Interest, and Related Arithmetic},
      howpublished = {Cryptology ePrint Archive, Paper 2004/034},
      year = {2004},
      note = {\url{}},
      url = {}
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