How To Prove It – Exercise 0.5

Oct 05 2012

Solutions to Exercises in the Introduction of How To Prove It by Daniel J Velleman.

Problem (5): Use the table in Figure 1 and the discussion on Page 5 to find two more perfect numbers.

Solution:

Euclid proved that if $2^n-1$ is a prime, then $2^{n-1}(2^n-1)$ is a perfect number.

From the given table, we will take two numbers such that $2^n-1$ is a prime number: 5 and 7.

When $n = 5$ :

$2^{n-1}(2^n-1) = 2^4(2^5-1)$ = 496, which is our first perfect number.

Similarly when n = 7, we get the next perfect number as 8128.

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