Q. Is 33,550,336 a perfect number

Yes, 33,550,336 is indeed a perfect number—specifically, it's the fifth perfect number ever discovered. A perfect number is defined as a positive integer that equals the sum of its proper divisors (all divisors excluding the number itself). For 33,550,336, when you add up all its divisors from 1 through 16,775,168, they sum exactly to 33,550,336. This number is intrinsically connected to Mersenne primes: it equals 2¹² × (2¹³ - 1), where 2¹³ - 1 = 8191 is itself a Mersenne prime.

Perfect numbers follow a remarkable pattern discovered by Euclid and later proven by Euler: all even perfect numbers have the form 2^(p-1) × (2^p - 1), where 2^p - 1 is a Mersenne prime. For 33,550,336, p = 13, making it directly tied to the mathematical properties of powers of 2 and prime numbers. Only a handful of perfect numbers are known—as of recent counts, fewer than 52—because they're extraordinarily rare and require finding Mersenne primes, which themselves are sparse. The first few perfect numbers are 6, 28, 496, 8,128, and then 33,550,336, showing massive gaps between successive perfect numbers. Whether odd perfect numbers exist remains one of mathematics' oldest unsolved problems, with no odd perfect numbers ever found despite extensive searching.