O'Reilly logo

Fearless Symmetry by Robert Gross, Avner Ash

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

74 CHAPTER 7
numbers from 0 to 6, you will find that x
2
5 0(mod7)has
no solutions, and therefore
5
7
=−1. And, finally, if you try
all seven possibilities in the congruence x
2
10 0(mod7),
again you will find that there are no solutions, so
10
7
=−1.
And that checks (7.1): 1 ·−1 =−1.
We now return to our polynomial x
2
+ 1, which corresponds
to choosing a =−1. In terms of Legendre’s notation, we have
discovered that
1
p
=
1ifp 1(mod4)
1ifp 3(mod4).
Before we continue, remember that if a is not divisible by p,then
a
p
is not 0. In particular,
1
p
is never 0.
When Is 2 a Square mod p?
The next number to consider is 2. We present you with the following
challenge if y ou like computer programming—otherwise, just read
our solution:
EXERCISE: List all of the odd primes under 100, and see for
which p you have
2
p
=−1, and for which p you have
2
p
= 1.
SOLUTION: It turns out that
2
p
= 1ifp = 7, 17, 23, 31, 41,
47, 71, 73, 79, 89, and 97. On the other hand,
2
p
=−1if
p = 3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, and 83.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required