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Fearless Symmetry by Robert Gross, Avner Ash

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QUADRATIC RECIPROCITY 75
You could stare at this list for a long time before discovering the
pattern, even when you know there is a pattern. If you did not know
there was something going on, you would have an even harder time
seeing the pattern. The answer is:
2
p
=
1ifp 1(mod8)
1ifp 3(mod8)
1ifp 5(mod8)
1ifp 7(mod8).
What this equation means is that if you have an odd prime p,
and you want to know whether or not 2 has two square roots in
F
p
, then you see what p is congruent to modulo 8, and look at the
corresponding line on the right-hand side of the equation to get
the value o f the Legendre symbol
2
p
.Inotherwords,
2
p
= 1
if p 1 or 7 (mod 8) and
2
p
=−1ifp 3 or 5 (mod 8). For
example, 47 7 (mod 8), and sure enough, 7
2
2 (mod 47) and
40
2
2 (mod 47).
When Is 3 a Square mod p?
We now move on even further to a = 3, and try again:
EXERCISE: Use the list of odd primes under 100 and decide
for which of those primes the equation
3
p
= 1 holds, and for
which you have
3
p
=−1.
SOLUTION: This time, our computer tells us that
3
p
= 1
when p = 11, 13, 23, 37, 47, 59, 61, 71, 73, 83, and 97, and
3
p
=−1whenp = 5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79,
and 89.

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