Math courses, and math contests for that matter, are often presented just a series
of questions, where it's your goal to try to come up with the answers. But is
mathematics, at its core, really just about answering questions? A lot of times
instead of being given a concrete question to answer, you're rather just
given some context to explore, maybe some numerical evidence to look at, and then,
in light of that numerical evidence, it'll be your job to come up with
interesting questions, interesting conjectures. Well, here's the result of
some numerical exploration. 2⁵ + 2 ≡ 1 mod 3; 3⁵ + 3 ≡ 1 mod 5;
3⁵ + 3 is also 1 mod 7; 2⁵ + 2 ≡ 1 modulo 11; 4⁵ + 4 ≡ 1 modulo 13;
7⁵ + 7 ≡ 1 modulo 17; and so on so. When you're confronted with this kind of
numerical evidence, what sort of patterns do you notice? Well
I noticed that 3, 5, 7, 11, 13, 17, 19, 23, these are all prime numbers, and over
here on the left hand side, I've got a number to the fifth power plus that same
number is congruent to one. So one sort of common pattern that I'm seeing here
is that I've got a bunch of data that fits into this sort of shape:
x⁵ + x is congruent to one modulo some prime. Is there a conjecture
that you could deduce from this? Sure. Conjecture: for every prime p there's an
integer x so that x⁵ + x is congruent to one modulo p. This is the
kind of conjecture that you might think is true when you're shown numerical
evidence like this. This is a bunch of examples of primes, 3, 5, 7, 11, and so on,
with x's, with integers, having the property that x⁵ + x is
congruent to one modulo the prime. Is that conjecture true? Well, if it is true,
it's not clear how we try to prove it. But anytime that we're given a general
statement, and we're trying to explore whether or not this general statement is
true or false, one good idea is to take that general statement, and try to think
about a special case. In this situation the
general statement involves a prime p so instead of trying to think about a prime
p in general, we could think about the special case where that prime p is 2. So
let's specialize the situation where p is equal to 2, so we're trying to find an
integer x with the property that x⁵ + x is congruent to 1
modulo 2. This would be the statement of the conjecture in a situation where that
prime p is 2. We're looking for an integer x so that x⁵ + x
is congruent to 1 modulo 2. In this case, well, this is really a statement about
whether or not something is even or odd, right, working modulo two, so it's maybe
natural to break this into cases as to whether or not x is even or odd. So let's
think about the situation where x is even, and in this case x is say 0 modulo
2, x⁵ is also even if x is even, and that means that x⁵ + x
is an even number plus an even number which is even. Well that's bad
news, so if x is even it can't be that x⁵ + x is 1 mod 2. x⁵ + x is even.
Okay, so let's think about the case where x is odd, well, in
this case, x⁵ is also odd, and then we've got x⁵ + x is
an odd number plus an odd number, which is necessarily even. So regardless of
whether x is even or odd, x⁵ + x ends up being even. And that tells
us that there isn't an integer with the property that x⁵ + x is 1
modulo 2, which means this conjecture, that looks so good based on the data
that we had, is in fact, wrong. So the conjecture as stated is false,
alright, it's not true when p is equal to 2, but that doesn't mean that we just
give up! Mathematics is less about answering the questions that were given,
and more about having a dialog, and what that means in this case is that instead
of just throwing away our false conjecture, we should try to salvage it.
Well this conjecture stated is is wrong, right, I mean we
noticed that when p was - this wasn't true because there isn't an integer so
that x⁵ + x is 1 mod 2, but what I mean by salvage is that I
want to fix this conjecture now in light of that counterexample to something
that might actually be true. So I'll just fix it this way: I'll avoid the counterexample
by adding an extra hypothesis that avoids the situation that we knew
was false. Maybe this is a true conjecture: is it the case that, for every
prime p ≠ 2 (because we know that one doesn't work) but for every prime p not
two there exists an integer x so that x⁵ + x is 1 modulo p? That's
a question, and that's a question for you. Is our salvaged conjecture even true?
That's for you to figure out. In the words of Glenn Stevens, "First you answer the
questions, then you question the answers." And that's really the spirit of what
we're trying to do here. All right, I don't want you just to be able to take
propositions, and put them in the true bucket and into the false bucket, right,
rather, I want you to be thinking of your own questions, coming up with your own
conjectures, and ultimately making your own discoveries.