#math trivia for #February13: 44 is the fifth number of the form p^2*q for distinct primes p, q. What are p and q? What were the first four?
— Burt Kaliski Jr. (@modulomathy) February 13, 2012
Factoring the day’s number, 44, quickly yields the values of p and q:
44 = 2^2 * 11 ,
giving p = 2, q = 11.
Because p and q are distinct, it follows that three of the first four numbers of this form involve p = 2 and the three odd primes less than 11:
12 = 2^2 * 3
20 = 2^2 * 5
28 = 2^2 * 7 .
The remaining number must involve p = 3 and q = 2:
18 = 3^2 * 2 .
As a final check that 44 is indeed the fifth such number, consider that the next number with p = 3 is 45, and the first with p = 5 is 50 — so we’ve covered all the possibilities less than 44.
Bonus question: How many other numbers of the form p2q will occur this year (i.e., up to 366)?
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