I am really confused about this example from my textbook. Here it is:
Suppose that we start with the universe that comprises only the 13 integers 2, 4, 6, 8,..., 24, 26. Then we can establish this statement:
For all n (meaning n = 2, 4, 6,..., 26), we can write n as the sum of at most three perfect squares.
The results in this table provide a case-by case verification showing the given (quantified) statement to be true. (We might call this statement a theorem.):
(Imagine these are in a table with 3 columns - I wasn't sure how to form tables in this)
2 = 1 + 1
4 = 4
6 = 4 + 1 + 1
8 = 4 + 4
10 = 9 + 1
12 = 4 + 4 + 4
14 = 9 + 4 + 1
16 = 16
18 = 16 + 1 + 1
20 = 16 + 4
22 = 9 + 9 + 4
24 = 16 + 4 + 4
26 = 25 + 1
This exhaustive listing is an example of a proof using the technique we call, rather appropriately, the method of exhaustion. This method is reasonable when we are dealing with a fairly small universe. If we are confronted with a situation in which the universe is larger but within the range of a computer that is available to us, then we might write a program to check all of the individual cases.
(Note that for certain cases in the table, more than one answer may be possible. For example, we could have written 18 = 9 + 9 and 26 = 16 + 9 + 1. But this is all right. We were told that each positive even integer less than or equal to 26 could be written as the sum of one, two, or three perfect squares. We were not told that each such representation had to be unique, so more than one possibility could occur. What we had to check in each case was that there was at least one possibility.)
My question: Why does it stop at 26 and not 28?
By looking at the tables, I am pretty sure it has something to do with perfect squares. I can't figure out a way that 28 can be figured by using the numbers 25, 16, 9, 4, and 1.
But, I'm not sure how to explain this with using correct mathematical terms. Am I on the right track? Can anyone shed some light on this please?
28 is indeed the first even natural number that cannot be represented as the sum of at most three perfect squares. However, this has nothing to do with the statement in the OP. The statement makes no claim about 28, either positive or negative, that is, that 28 can or cannot be represented as the sum of at most three perfect squares. As I said, the universe includes even numbers up to 26 by definition. Second-guessing a definition is not a mathematical activity.