Talk:Quadratic reciprocity

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Dear DYLAN LENNON,

Could you please explain how you intend to use quadratic reciprocity to prove fermat's theorem on sums of two squares. The only part that I can see is vaguely relevant is the first supplementary theorem, i.e. that \left(\frac{-1}p\right) = (-1)^{(p-1)/2}, which is by far the most trivial part of QR. (I have taken the liberty of reverting your edit again until you can provide an explanation.) Dmharvey 01:53, 5 February 2006 (UTC)

You can learn by reading this note. (http://www.math.nmsu.edu/~history/book/numbertheory.pdf) Good luck DYLAN LENNON 02:45, 5 February 2006 (UTC)

I've had a look, and I can't find what you mean. Could you give a page number perhaps? Even better, which paragraph/sentence supports your claim? Dmharvey 02:50, 5 February 2006 (UTC)

Contents

[edit] ZX81

There is this lovely line in the ZX81 manual:

65537 is a Fermat prime, 216 + 1. Use this, and Gauss's Law of Quadratic Reciprocity, to prove that 75 is a primitive root modulo 65537.

Needless to say I am none the wiser!

Having now discovered the primitive root modulo n page, it transpires that all I needed to do was to verify that 75^{2^{15}} \not\equiv 1 \pmod{65537}

[edit] n-th reciprocity

Is there a further formulation of reciprocity?? let's say an study of the solutions:

xn = pmod(q) xn = qmod(p) —The preceding unsigned comment was added by 85.85.100.144 (talk) 21:01, 19 February 2007 (UTC).

[edit] Python code for residue table

I started trying to make a table of residues to illustrate quadratic reciprocity, but it soon got very painful to do by hand. So I wrote a Python script (my first!) to do it for me. Of course, just editing the script here won't update the table, you'll have to run it on your own machine :-)


    # find primes from 3 up to max
    max = 50
    primes = []
    for n in range(3, max):
        composite = False
        for d in range(2, n-1):
            if n % d == 0:
                composite = True
                break
        if not composite:
            primes.append(n)
            
    count = len(primes)

    yes_marker = '✓'   # tick (U.S. "check") for residues
    no_marker  = '✗'   # cross for non-residues

    def colortag(n):
        if n % 4 == 1:
            return 'bgcolor=#e0ffff'
        else:
            return 'bgcolor=#ffe0e0'

    # computes Legendre symbol (a/q)
    # assumes a and q positive, q prime, (a, q) = 1
    def legendre(a, q):
        for n in range(1, q-1):
            if (n * n) % q == a % q:
                return 1;
        return -1;
        
    # print table header
    print '{| class="wikitable"'
    print '|-'
    print '| || colspan=' + str(count+1), 'align="center" |', "''p''"
    print '|-'
    print '| rowspan=' + str(count+1), "|  ''q''  || ",
    for p in primes:
        print '||', colortag(p), 'align="center" style="border-bottom:2px solid" |', "'''" + str(p) + "'''", 
    print

    # now the main table
    for q in primes:
        # first column
        print '|-'
        print '|', colortag(q), 'align="right" style="border-right:2px solid" |', "''' " + str(q) + " '''",

        # remaining columns
        for p in primes:
            print '||', colortag(1+(p-1)*(q-1)/2), '|',

            if p == q:
                print ' ',
            else:
                # symbol for (p/q)
                if legendre(p, q) == 1:
                    print yes_marker,
                else:
                    print no_marker,

                if legendre(q, p) == 1:
                    print yes_marker,
                else:
                    print no_marker,
        print
            
    print '|}'

Dmharvey 03:22, 21 April 2006 (UTC)

[edit] Chart

Something's wrong with the chart... the check and cross marks both look like boxes.

63.228.45.224 16:23, 28 May 2007 (UTC)

I changed the table to use images instead of Unicode characters, as the Unicode characters don't show up on all computers (see previous comment). I put the new code on User:chridd/sandbox1 because of the Don't edit comments on talk pages policy. I changed

    yes_marker = '✓'   # tick (U.S. "check") for residues
    no_marker  = '✗'   # cross for non-residues

to

    yes_marker = '[[Image:Yes check.svg|10px]]'   # tick (U.S. "check") for residues
    no_marker  = '[[Image:Black x.svg|10px]]'   # cross for non-residues

~User:chridd [[tʃɹɪ|Special:Contributions]] 03:02, 20 September 2007 (UTC)

[edit] history section

I will add a brief history section soon. User:Virginia-American/Sandbox has changes for the article on quadratic residues. I don't anticipate anything so extensive here.

How about FAQ's

Why did CFG do so many proofs? Why has everyone else as well? Why "law" (it's not a thing like other laws, eg. commutative law of addition)? Virginia-American (talk) 01:05, 28 February 2008 (UTC)