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Fearless Symmetry by Robert Gross, Avner Ash

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RECIPROCITY LAWS 197
by means of the Frobenius elements, will match the output from
one of a certain set of black boxes (those with the correct label).
Sometimes, reciprocity laws are theorems that have been proven.
Sometimes they are only conjectured and have not been proven yet.
And, of course, you can mistakenly conjecture a false reciprocity
law and find a numerical disproof. But the reciprocity laws in the
remainder of this book are all either theorems or conjectures that
most experts would say are probably going to turn out to become
theorems.
Digression: Conjecture
The word conjecture means “guess. The way it is used in math-
ematics is “educated guess. In this digression, we will mention
several conjectures from parts of mathematics other than Galois
theory.
There are at least two different classes of great conjectures:
1. Those based on evidence.
2. Those based on analogy.
Some conjectures that are currently the object of intense study by
mathematicians include the Poincar
´
e Conjecture (which reportedly
may have been proven true as of this writing), the Riemann
Hypothesis (which is a conjecture, though it is called a “hypothe-
sis”), and the conjecture that P = NP.” As a matter of fact, there is
a million-dollar prize (offered by the Clay Mathematics Institute)
for the first resolution of any of these three conjectures, as well as
some others.
2
Mathematicians may not increase their efforts be-
cause of the prize, but public interest has certainly increased.
Sometimes, serious conjectures are proven false. An example is
the Mertens Conjecture. First define the M
¨
obius function µ(n) for
positive integers n: µ(n) = 0ifn is divisible by a square number
larger than 1; otherwise, µ(n) = 1ifn has an even number of prime
factors and µ(n) =−1ifn has an odd number of prime factors. Then
2
A discussion of the Millennium problems may be found in (Devlin, 2002).
198 CHAPTER 17
you define the Mertens function for x > 1by
M(x) = µ(1) + µ(2) + µ(3) +···+µ(n),
where n is the largest integer less than or equal to x. The Mertens
Conjecture states that for all x > 1,
|M(x)| <
x.
There is a known way to prove the Riemann Hypothesis from this
statement, if it were true. If you work out M(x) for small values of x,
the conjecture seems to be true. In 1985, however, Andrew Odlyzko
and Herman te Riele proved that the Mertens Conjecture is false.
Fermat’s Last Theorem states: If x, y, and z are all nonzero
integers, and n is an integer greater than 2, then it cannot happen
that x
n
+ y
n
= z
n
. Before this theorem was proved by Andrew Wiles,
it was a conjecture that fell in both classes, which is why it
was regarded as particularly strong. In the simplest sense, there
was evidence supporting it, as no one ever found integers solving
x
n
+ y
n
= z
n
. The supporting evidence in fact was much stronger:
For many particular values of n, mathematicians, starting in the
eighteenth century, had proved that there were no solutions of
x
n
+ y
n
= z
n
.
There was also at least one analogy that helped convince many
mathematicians that Fermat’s Last Theorem was true. There are
many statements about integers that can also be made about
polynomials, and in general the true statements about integers
tend also to be true of polynomials. Proving things about polyno-
mials is usually much simpler than proving things about integers,
for at least two reasons: Polynomials have roots, and they can
be differentiated. Using these tools, it is not all that difficult to
show that there are no nonconstant polynomials with complex
coefficients f (x), g(x), and h(x) so that f (x)
n
+ g(x)
n
= h(x)
n
if n > 2.
Another conjecture that is strongly supported by evidence in both
categories is called the Riemann Hypothesis. This is the statement
that if the real part of the complex number s is positive, then the
Riemann ζ -function ζ (s) is zero only when s =
1
2
+ it (where t is a
real number). On the one hand, you can do numerical computations
and the Riemann Hypothesis looks true. On the other hand, there

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