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#math trivia for #June17: #169 is one of eight day-numbers that are squares of primes. What are the others?

— Burt Kaliski Jr. (@modulomathy) June 18, 2012

Recall that “day-numbers” are integers between 1 and 366 — the numbers of the days of the year (366 is included because the problem was given during a leap year, though that won’t affect the result).

The largest square between 1 and 366 is 19² = 361, so all the answers must be squares of a prime less than or equal to 19.  There are eight such primes, and consequently eight acceptable day-numbers:

• 1 = 1²
• 4 = 2²
• 9 = 3²
• 25 = 5²
• 49 = 7²
• 121 = 11²
• 169 = 13²
• 289 = 17²
• 361 = 19²

Note:  The number of prime squares less than or equal to a limit N is approximately 2 sqrt(N) / ln(N), where ln is the natural logarithm.  For N = 366, the value is about 6.48, which is reasonably close to the actual answer 8.  The estimate is a consequence of the Prime Number Theorem, which states that the number of primes less than or equal to a limit N is approximately N / ln(N).  Because we’re looking for prime squares, our estimate is based on the number of primes less than or equal to the square root of the limit, which is sqrt(N) / ln(sqrt(N)) = sqrt(N) / (ln(N) / 2) = 2 sqrt(N) / ln(N).