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[Untitled]
- Valid proof?::
αi(p−1)/2 ≡ 1 (mod p). By Fermat's little theorem, p − 1 divides i(p − 1)/2
How does this follow?
- —Preceding unsigned comment added by 86.131.60.104 (talk) 21:54, 6 February 2011 (UTC)
- I think it follows not from Fermat's little thorem but from the definition of primitive roots: p-1 is the minimum k such that , so if k should be a multiple of p-1.222.148.64.59 (talk) 14:34, 2 May 2011 (UTC)
I think the example could be improved: right now it doesn't illustrate the criterion. We should start with some number a, show that it is a quadratic residue because it can be written as a square, and then compute its (p-1)/2 power to show that the result is 1. Then use a nonresidue and show that you get -1. AxelBoldt 03:07 Oct 16, 2002 (UTC)
- Keep in mind that this particular example took me 4-5 hours to complete it right. The example for a=17 first of all tries to show how we can figure it out which numbers are squares modulo a prime without using Euler's criterion for some small a. Then we can watch:
- k2 ≡ 17 (mod p).
- We get all p which solves it {2,4,8,16}. Now using Euler's criterion we see that:
- 17(p-1)/2 ≡ 1 mod p; ,
- for all p from {2,4,8,16}. We can also see that the latter is true for all p from {9,13,19,43,59,...}. --XJamRastafire 10:50 Oct 16, 2002 (UTC)
The second part of "Proof of Euler's criterion" is not complete. It shows that if a is not a q.r. modulo p then a^((p-1)/2) is not 1, but it doesn't prove that it is -1.
I have made a major cleansweep of this article. The first section really needed attention because the line of reasoning was somewhat unclear - this seems to be the case for most proofs of this criterion found on the net....most of them just cites Fermats little theorem like it was some magic wand. I hope the line of reasoning has become clear now. Holretz
Jump in math
The very last example makes sense if you already know the criterion, but not if you do not. It claims 38=16. That is not true. 38=6561. Then 6561%17=16. I assume that use of % for mod in this article is not proper. What is the proper format for representing 6561%17 so that users who are attempting to understand the criterion will be able to follow this current missing step? -- kainaw™ 02:22, 8 March 2009 (UTC)
Eulers Criterion
shouldn't this be "Eulers criterion?" —Preceding unsigned comment added by 69.181.220.209 (talk) 21:09, 4 July 2010 (UTC)
- No, why? It's a perfectly regular use of a possessive apostrophe.—Emil J. 12:38, 5 July 2010 (UTC)
Generalized Euler's criterion
We have been taught that the following is called the Generalized Euler's criterion:
- Let N > 1 have a primitive root, and a co-prime to N. Then xk ≡ a (mod N) has a solution iff .
Now, I don't know whether this is the correct name of the above statement (so I'm not editing the article), but this article mentions neither the statement itself nor contains a reference to a place that does. At least the latter should be done. 08:46, 9 August 2012 (UTC) — Preceding unsigned comment added by Ybungalobill (talk • contribs)
- I've never heard it called the generalized Euler's criterion. = Virginia-American (talk) 12:22, 9 August 2012 (UTC)