A formula related to multiplicative order

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Discussion Overview

The discussion revolves around a proposed formula related to the multiplicative order of elements in a finite field GF(p). Participants explore the validity and implications of the formula, its components, and its potential significance in mathematical and cryptographic contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Post 1 introduces a formula involving a summation over elements in GF(p) and proposes a relationship with the Mobius function and Euler’s-Totient function.
  • Post 2 expresses uncertainty about the novelty of the formula due to lack of responses.
  • Post 3 challenges the validity of the proposed formula, stating that the left-hand side (LHS) and right-hand side (RHS) cannot be equated due to differing domains.
  • Post 4 seeks clarification on the correctness of the mathematical expression and its significance, presenting an alternative formulation.
  • Post 5 argues against the arbitrary coercion of elements from GF(p) into the integers, highlighting the lack of a canonical choice in such transformations.
  • Post 6 acknowledges the potential error in the initial equation and expresses a desire for guidance on publishing the results, while also proposing a related result that may have cryptographic applications.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the proposed formula. There are competing views regarding the correctness of the mathematical expressions and their implications.

Contextual Notes

Participants express uncertainty regarding the definitions and transformations involved in the proposed formula, particularly the transition between finite fields and integers. The discussion also highlights unresolved mathematical steps and the implications of the proposed results.

Who May Find This Useful

Mathematicians and researchers interested in number theory, finite fields, and cryptography may find this discussion relevant.

praneeth
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I want to submit a paper on proof of this formula, so can some one please tell whether this already exists or not?

Let S’(A) be the set of elements in GF(p) such that S’(A) = {x/ O(x,p) = A}. Here A should be
the factor of (p-1) and A>2, where p is prime, then
∑x = μ(A) + ½* T(A)*p;
where the summation is over all the elements of set S’(A) and
O(x,p) : Order of x with respect to p, (by order it is meant to be multiplicative order).
μ(A) : Mobius function of A.
T(A) : Euler’s-Totient function of A.
 
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no replies which means I can write a paper on proof of this theorem. Is that it??
 
It means no one has looked at it, or thought about it, not that it is new or novel. Not that this is necessarily the correct place to even ask such a question.

The most obvious problem with what you wrote is that it does not and cannot make sense. You have an equality. The LHS is a sum of elements in a finite field, the right hand side is an integer.
 
1) If you look at it as a mathematical expression, is that correct?? addition of elements taken from a finite field in the integer domain (changing the domain).
(or)
2) (∑x - μ(A))mod p = 0
is that correct? I just want to know if it has got any significance?
 
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You cannot arbitrarily coerce elements from GF(p) into Z, as you did. There is no canonical choice - should I pick -1 or p-1, for instance?

You can of course, take an integer like u(A), and reduce it mod p, if you wish.

Of course, since you're in GF(p), the multiplicative group is cyclic, so one has a nice description of what the elements are with a given order, and how many of them there are etc. Try Le Veque for more information.
 
Mr. Grime, So, the 1st equation is wrong. I don't know "Le Veque". If its new what should I do with it. Can you suggest anyone journal to write to about this? because every journal I saw won't accept new proofs to theorems as its articles.

Also I want to mention that
{∑(x^r)-μ(A)} mod p=0; for any r co-prime to A. This result might be useful in cryptography.
 
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