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Fearless Symmetry by Robert Gross, Avner Ash

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QUADRATIC RECIPROCITY 73
possible to do a lot of mathematics that would otherwise be tedious
or too complicated to comprehend when stated in words.
Another example is powers. We write x
2
to mean x times x,
without g iving it a second thought. But as we write this, we can
think of the exponent as another variable, and write x
n
.Wecan
write down the power law of exponents: x
n
x
m
= x
n+m
. Influenced by
this symbolism, we can then start thinking about x
n
where n is no
longer an integer. We can write down x
1/2
and give a meaning to
that symbol: x
1/2
means the square root of x.
A fascinating example of the advantages and disadvantages
of notation occurred when calculus was invented (or discovered).
Newton used one symbol to stand for the process of differentiation;
Leibniz used a different symbol. Newton w as perhaps the greater
mathematician and scientist, but his notation was quite hard to
use. Leibniz’s notation was so good that every year, thousands
of students compute derivatives without ever fully understanding
them, because the notation almost forces the students to do the
right computation—at least most of the time!
We will see immediately that Legendre’s notation cleverly helps
us to notice and remember a certain multiplicative structure more
easily than we would have with the na
¨
ıve “unshifted” notation.
Multiplicativity of the Legendre Symbol
Legendre’s notation helps clarify what happens when p is fixed and
the number a varies. In particular, after playing around with some
examples, you may discover—and it is possible to prove—that
a
p

b
p
=
ab
p
. (7.1)
EXERCISE: Show that
2
7

5
7
=
10
7
.
SOLUTION: You can check that 3
2
2 0(mod7)and
4
2
2 0 (mod 7). Therefore,
2
7
= 1. If you try the seven

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