Week 9

Example 1. Express the number 51 as the difference of squares of the numbers.

Of course, you can find the solution just below, but it is highly recommended that, you first try to solve it on your own.

Just remember the words of Paul Halmos, who says “the only way to learn mathematics is to do mathematics”.

Solution 1. We know that (x + 1)2 = x2 + 2x + 1 and (x − 1)2 = x2 − 2x + 1. Therefore,

(x + 1)2 − (x − 1)2 = (x2 + 2x + 1) − (x2 − 2x + 1) = 4x.

Since 4 = 22, above relation can be re-written as

Substituting x = 51 in the last relation, it follows that

Note. Of course, there exist the trivial solution 51 = 102 − 72.

Solution 2. Evidently, 3 < x. Again, from the given information we have

This relation helps us to find a bound on y. Clearly, 5y < 19 and the nearest integer satisfying this is y = 3. Therefore,

It follows that, x = 7. Therefore, (x, y) = (7, 3).

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: