Talk:Quadratic reciprocity

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 $\backslash left(\backslash frac\{-1\}p\backslash right)\; =\; (-1)^\{(p-1)/2\}$, 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)

There is this lovely line in the ZX81 manual:

:65537 is a Fermat prime, $2^\{16\}+1$. 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 $75^\{2^\{15\}\}\; \backslash not\backslash equiv\; 1\; \backslash pmod\{65537\}$

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

$x^\{n\}=p\; mod\; (q)$ $x^\{n\}=q\; mod\; (p)$ —The preceding unsigned comment was added by 85.85.100.144 (talk) 21:01, 19 February 2007 (UTC).

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)

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 = '10px'   # tick (U.S. "check") for residues
no_marker  = '10px'   # cross for non-residues

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

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)

I colorized one of the tables and put the border back around it. Anyone have ideas for inproving the aesthetics of this? thanks Virginia-American (talk) 02:52, 12 November 2008 (UTC)

Does anyone know anything about this? Since the LS relation reduces to a Guass sum, and QR can be easily proved using Gauss sums, is this the extent of it, or is there more? —Preceding unsigned comment added by Virginia-American (talk • contribs)

This article is very nice and has a lot of good content, but the beginning reads more like an exposition than an encyclopedia article. There needs to be statement of the full theorem (or at least one version of the theorem) much earlier in the article, ideally in the introduction or near the beginning of the first section. The tables of numbers and such are helpful for motivation and understanding, but they ought to be put in a "Motivation" section that occurs after the statement of the law. Jim (talk) 20:25, 23 November 2008 (UTC)

: done Virginia-American (talk) 14:28, 24 November 2008 (UTC)

"A number of proofs of the theorem, especially those based on Gauss sums derive this formula.[20] or the splitting of primes in algebraic number fields,[21]"

Description in one word: Ungrammatical. — Preceding unsigned comment added by 189.63.168.238 (talk) 01:45, 16 April 2016 (UTC)

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In the section Supplements to Quadratic Reciprocity there is a text "The question becomes more interesting for −1. Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47.". I didn't manage to see the referred table neither above nor below. 77.219.9.21 (talk) 23:27, 25 February 2025 (UTC)