Question

A number of the form $2^{(p-1)}(2^p-1)$ is an even perfect number if 'p' and $2^{(p-1)}$ are prime numbers.

• A. True
• B. False

Answer 1

The correct option is A True

Euclid proved that 2^p−1(2^p − 1) is an even perfect number whenever 2^p − 1 is prime. For example, the first four perfect numbers generated by the formula 2^p−1(2^p − 1), with p a prime number are as follows:

For p = 2: 2¹(2² − 1) = 2 x 3 = 6
For p = 3: 2²(2³ − 1) = 4 x 7 =28
For p = 5: 2⁴(2⁵ − 1) = 16 x 31 = 496
For p = 7: 2⁶(2⁷ − 1) = 64 x 127 = 8128