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).