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How To Prove It by Velleman

Prove It by Velleman, Introduction, Problem 5

August 7, 2014 etoix Leave a comment

Problem

Use the table in Figure 1 and the discussion on p. 5 to find two more perfect numbers.

Solution

Page 5 tells us that if $2^{n} - 1$ is prime, then $2^{n-1} ( 2^{n} - 1 )$ is perfect.

Figure 1 shows that $2^{n} - 1$ is prime for $n = 5$ and $= 7$ .

By calculator:

$2^{5-1} ( 2^{5} - 1) = 496$

$2^{7-1} ( 2^{7} - 1) = 8128$

Both 496 and 8128 are perfect numbers.

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