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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "#quadresinvestigatio
n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#Duane A. Cooper, Dept
. of Mathematics, Maple Workshop, 10/8/02" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "#Here's a quick investigati
on to inspire conjectures for what primes " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 56 "#2 is a quadratic residue and for what primes it is
 not." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "#First let's recall what a quadratic resi
due is." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "#Pick a prime, a
ny prime (well, keep it small for now)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "p:=11;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"#6" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 68 "#Next let's look at lists of the integers and \+
their squares, 0..p-1," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "#
and then we'll examine those squares modulo p." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"L := [seq(i,i=0..p-1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7-\"
\"!\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "L := [seq(i^2,i=0..p-1)];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"LG7-\"\"!\"\"\"\"\"%\"\"*\"#;\"#D\"#O\"#
\\\"#k\"#\")\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "L := \+
[seq(i^2 mod p,i=0..p-1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7-
\"\"!\"\"\"\"\"%\"\"*\"\"&\"\"$F+F*F)F(F'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 66 "#Any number n in the latter list is said to be a qu
adratic residue" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "#mod p b
ecause the congruence x^2 == n mod p has a solution." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 64 "#Any number k, where 0 <= k < p, that is \+
not in the list is said" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "
#to be a quadratic nonresidue mod p; x^2 == k mod p has no solution." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 66 "#Now let's invoke Maple's number theory package \+
so that we can use" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#the \+
Legendre symbol.  Recall that for odd prime p," }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 52 "#legendre(a,p)=0 if a is a multiple of p; othe
rwise," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "#legendre(a,p)=1 \+
if a is a quadratic residue of p, and" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "#legendre(a,p)=-1 if a is a quadratic nonresidue of p
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "#For example, legendre
(1,3)=1; legendre(2,3)=-1; legendre(3,3)=0." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "wit
h(numtheory):" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, the protected n
ame order has been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "legendre(3,p);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "legendre(2,
p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 64 "#Now let's use our computational power to evalua
te legendre(2,p)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "#for al
l the primes p < 100 (or however high you wish to explore)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 99 "for i from 2 by 1 while ithprime(i) < 100 do\n   pr
int(ithprime(i), legendre(2,ithprime(i)))\nend do;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"(\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#8!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#<\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#>!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#B\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#H!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#J\"\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#P!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#T\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#V!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#Z\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#`!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#f!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#h!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#n!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#r\"\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#t\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#z\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#$)!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#*)\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#(*\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "#Need a big ol' juicy hint?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 60 "#If so, I'll provide the value of HINT for you to use below." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "hint:=HINT;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%hintG%%HINTG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "for i fr
om 2 by 1 while ithprime(i) < 100 do\n   print(ithprime(i), legendre(2
,ithprime(i)), ithprime(i) mod hint)\nend do;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$!\"\"-%%modpG
6$F#%%HINTG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&!\"\"-%%modpG6$F#%
%HINTG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(\"\"\"-%%modpG6$F#%%HIN
TG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#6!\"\"-%%modpG6$F#%%HINTG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8!\"\"-%%modpG6$F#%%HINTG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6%\"#<\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6%\"#>!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6%\"#B\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#H!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#J\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#P!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#T\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#V!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#Z\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#`!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#f!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#h!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#n!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#r\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#t\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#z\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#$)!\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#*)\"\"\"-%%modpG6$F#%%HINTG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%\"#(*\"\"\"-%%modpG6$F#%%HINTG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "#Now prove it!" }}}}{MARK "41 0 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }