Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Spring 2004 MATH025.1001
Proofs in mathematics
We are used to believe, or to make guesses, as to whether something is true
or not.
In mathematics we try to find absolute truths, to the best of our ability.
Here are some examples to illustrate this process.
- We all know about the whole numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, and so on.
Simple experimenting shows that we can write all small numbers as sums of
four squares, sometimes in more than one way.
For example,
- 0 = 0+0+0+0
- 1 = 1+0+0+0
- 2 = 1+1+0+0
- 3 = 1+1+1+0
- 4 = 1+1+1+1 = 4+0+0+0
- 5 = 4+1+0+0
- 6 = 4+1+1+0
- 7 = 4+1+1+1
- 8 = 4+4+0+0
- 9 = 9+0+0+0 = 4+4+1+0
- blablabla
- 107 = 81+25+1+0 = 81+16+9+1 = 64+25+9+9
- blablabla
By checking many cases we may believe, but will not know,
that all numbers are sums of four squares.
In the eighteenth century the Frenchman Lagrange proved that all numbers
are sums of four squares.
Shortly thereafter, the Swiss Euler gave a nice proof which is the better known
one.
The proofs essentially show for all positive numbers n that if one already
knows that all numbers less than n are sums of four squares, then n itself is
also a sum of four squares.
We already saw that 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are sums of four squares.
So by their proof, 10 is also a sum of four squares.
Now we know that 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are sums of four squares.
So, again applying their proof, 11 is also a sum of four squares.
Now we know that 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 are sums of four
squares.
So, again applying their proof, 12 is also a sum of four squares.
And so on.
Thus, by mathematical induction, all numbers are sums of four squares.
- For illustration purposes, let us also look at a simpler property which
we can prove on our own.
Claim:
- The sum 1 + 2 + 3 + ... + (n-1) + n of the first n positive whole
numbers, equals n(n+1)/2
At first we may wish to verify this claim for a few small values, just to see
that it stands a chance.
- 1 = 1*2/2
- 1 + 2 = 3 = 2*3/2
- 1 + 2 + 3 = 6 = 3*4/2
- 1 + 2 + 3 + 4 = 10 = 4*5/2
- 1 + 2 + 3 + 4 + 5 = 15 = 5*6/2
- 1 + 2 + 3 + 4 + 5 + 6 = 21 = 6*7/2
Ok, so it seems to work.
Now we must prove it for all whole numbers.
- The claim holds when n equals 1. We tested that case above.
- Suppose that the claim has been proved for a certain value, say p.
So we suppose that 1 + 2 + 3 + ... + (p-1) + p equals p(p+1)/2.
Then 1 + 2 + 3 + ... + (p-1) + p + (p+1) equals p(p+1)/2 + (p+1), which
equals (p(p+1) + 2(p+1))/2, which simplifies to (p+2)(p+1)/2.
So 1 + 2 + 3 + ... + (p-1) + p + (p+1) equals (p+1)(p+2)/2.
Thus the claim also holds for p+1.
We call this the induction step.
- Now we claim that, by mathematical induction, the claim holds
for all n.
Let us see in a few glances why this principle of mathematical induction is
true.
Recall that we have shown the claim for n = 1, and that if the claim holds for
a value n = p, then it also holds for value n = p+1 (induction step).
As we wrote, the claim holds for n = 1.
So, by the induction step, the claim also holds for n = 2.
So, again by the induction step, the claim also holds for n = 3.
So, again by the induction step, the claim also holds for n = 4.
So, again by the induction step, the claim also holds for n = 5.
So, again by the induction step, the claim also holds for n = 6.
So, again by the induction step, the claim also holds for n = 7.
And so on.
The claim holds for all positive whole numbers.
- It is tempting to conclude that if we test a claim for sufficiently many
small values of n, say for n = 0, for n = 1, n = 2, n = 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, and 15 or so, that we could as well believe that the claim
is true for all n.
You may do so, but you do not know with mathematical certainty that your belief
is correct.
Consider the following example.
Recall that a whole number bigger than 1 is called prime if we can not write it
as a product of smaller whole numbers.
So 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on, are primes.
But 4 is not prime because 4 = 2*2; and 6 is not because 6 = 2*3; and 8 is not
because 8 = 2*2*2; and 9 is not because 9 = 3*3; and so on.
Consider the function
For all whole numbers n the function f returns another whole number.
For example
- f(0) = 0+0+41 = 41
- f(1) = 1+1+41 = 43
- f(2) = 4+2+41 = 47
- f(3) = 9+3+41 = 53
- f(4) = 16+4+41 = 61
- f(5) = 25+5+41 = 71
- f(6) = 36+6+41 = 83
- f(7) = 49+7+41 = 97
- f(8) = 64+8+41 = 113
- f(9) = 81+9+41 = 131
- f(10) = 100+10+41 = 151
- f(11) = 121+11+41 = 173
It is a tedious but easy exercise to verify that all these returned values are
prime.
It is not unreasonable to believe that the function f will always
return primes.
But we do not know.
To strengthen the belief, we may verify that f(12), f(13), f(14), f(15), f(16),
f(17), f(18), f(19), f(20), f(21), f(22), f(23), f(24), f(25), f(26), f(27),
f(28), f(29), f(30), f(31), f(32), f(33), f(34), f(35), f(36), f(37), f(38),
and f(39) are prime.
Still we do not know.
In fact the belief is incorrect, for
- f(40) = 40*40+40+41 = 1681 = 41*41
Last updated: January 2004
Comments & suggestions:
wimr@mscs.mu.edu