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Blast into Math! Prime nummers: indestructimle muilding mlocks

9. * Show that every even perfect number is of the form 2 p−1 q where p is prime and q is the
Mersenne prime
p
q =2 − 1.
Are there infinitely many even perfect numbers?

10. * What are some odd perfect numbers?
11. Look up some of the “mystical properties” believed to be held by perfect numbers.
12. There are several different correct proofs that there are infinitely many prime numbers.
13. This exercise is an opportunity for you to explore some different mathematical styles. Do

some research into different proofs that there are infinitely many primes and find your
favorite, whether it’s the proof in this book or another proof that you find on your own. What
do you like about your favorite proof? Are there some proofs that you don’t like? Why not?

5.6 Examples and hints

Remember to try the exercise problems before looking at the examples. You could surprise yourself and
solve them all without any help!

• Hint for #1: Use the FTA.
• Hint for # 2: If a number n ∈ N is not prime, then by the FTA, you can write n as a

product of prime numbers. This also means that n is divisible by at least one prime. So,
p
you can write n = pq , where is prime, and q ∈ N . What happens if both p and q are
√
greater than n ?
1
• Hint for #3: A Mersenne prime is one less than a power of two. 2 =2, and 2 − 1= 1 is
2
2
not prime. How about 2 ? 2 =4, and 4 − 1= 3 is prime. Now, continue with the powers
of 2 and see if when you subtract one, the number is a prime. Can you prove that you can
always find a larger Mersenne prime? For the twin primes, start by writing down the prime
numbers beginning with 2. The next prime is 3. But, 2 and 3 are only one apart. So they’re
not “twins.” The next prime is 5. What is the difference between 5 and the previous prime

(3)? Are they twins? Now you can continue checking for twins. Can you prove that you can
always find more “twins? ” It could be useful to look up Mersenne primes and twin primes
on the internet.
• Hint for #4: The sum of two consecutive prime numbers larger than 2 is the sum of two odd
numbers, because all primes (except 2) are odd. Think about this, and write a proof. So, the
sum of two consecutive primes is a composite number n. Think about the definition of a
composite nu mber n : it is not prime. This means, by definition of prime, that it is divisible
by some x ∈ N with 1 <x<n . By definition of divide, there is y ∈ N such that

n = xy.

Since x< n , and y> 1 you can prove #4 using induction and the proof of the FTA as a guide.

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