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#math trivia for #September4: #247 is the difference of two squares and the product of two primes. What are they and how are they related?

— Burt Kaliski Jr. (@modulomathy) September 5, 2013

Answer:  The squares are 256 and 9; the primes are 13 and 19. The relationship is as follows:

• 256 is the square of 16, and 16 is the average (half the sum) of 13 and 19

• 9 is the square of 3, and 3 is half the difference between 13 and 19

This is an instance of a general property: If a number n is the product of two primes p and q, i.e., n = pq, then it is also the difference between the squares of the average of the primes, i.e., (p+q)/2, and half their difference, i.e., (p–q)/2. In equation form:

((p+q)/2)² – ((p–q)/2)² = (p²+2pq+q²)/4 – (p²-2pq+q²)/4 = 4pq/4 = pq  .

Note: There’s one other way to express 247 as the difference of two squares: 247 = 134²-133². This is also an instance of a general property: If a number n is odd, then it is also the difference between the squares of (n+1)/2 and (n-1)/2. However, because 134 and 133 aren’t directly related to 13 and 19 (at least not as closely as 256 and 9 are), this other way isn’t part of the “full” answer.