#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).