Burt Kaliski Jr. (@modulomathy)
3/8/12 7:21 AM
#math trivia for #March8: #68 is both the sum of squares and the difference of squares. What are the squares?
The sum of squares is 64+4, or 8^2+2^2.
The difference is 324-256, or 18^2-16^2.
General rule for the difference of squares: if an integer N can be expressed as the product of integers A*B where A and B are either both even, or both odd, then N can also be expressed as the difference of squares
N = ((A+B)/2)^2 – ((A-B)/2)^2.
The reason that A and B must have the same parity is that otherwise (A+B)/2 and (A-B)/2 won’t be integers. The rule thus works for integers that are odd, or divisible by 4, but not for integers that are even but not divisible by 4 (such as 2 or 6). Those integers cannot be expressed as a difference of two squares, but the rest can. There may be more than one representation. For instance, 15 is 8^2-7^2, and 4^2-1^2, following two different factorizations.
In the present case, an equal-parity factorization of 68 is 34*2, leading to the squares shown.
The general rule for the sum of squares is a little more complicated, because it requires first that one express each factor of N as a sum of squares. An interesting recursive problem, which I look forward to pondering further.