#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 }= (*p*^{2}+2*pq*+*q*^{2})/4 – (*p*^{2}-2*pq*+*q*^{2})/4 = 4*pq*/4 = *pq .
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Note: There’s one other way to express 247 as the difference of two squares: 247 = 134^{2}-133^{2}. 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.