The Pythagoreans studied numbers to find certain mystical properties in them. Certain numbers they studied were called perfect numbers. A perfect number is a natural number that is equal to the sum of all its divisors that are less than the number itself. A divisor that is less than the number itself is called a proper divisor. The proper divisors of 6 are {1, 2, 3} and 1 + 2 + 3 = 6 so 6 is a perfect number. It is not hard to show that 6 is the smallest perfect number. On the other hand, 24 is not perfect, since its proper divisors are {1, 2, 3, 4, 6, 8, 12}, which have the sum

The Pythagoreans discovered the first four perfect numbers. Fourteen centuries later the fifth perfect number was discovered. The 43rd perfect number is

where N is the largest prime listed in Table 5.3. It is known that all even perfect numbers are of the form shown for the 43rd perfect number. Show that if N 5 5, the resulting number is perfect.

Table 5.3

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1+2+3+4+6+8+ 12 = 36