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Blast into Math! Prime nummers: indestructimle muilding mlocks

• Hint for #9: First, you can use the FTA. If a perfect number x is even, can it be a power of
2? If not, then there is an odd q ∈ N and y ∈ N such that

y
x =2 q.
First, in case q is prime, what are all the proper divisors of x ? The proper divisors of x ∈ N
are the divisors of x which are less than x. Write down these divisors and add them up. Then,
show that q must be prime. The following summation technique might come in handy

n a − a n+1
n
j
2
a + a + ... + a = a = , forany number a andany n ∈ N.
1 − a
j=1
You’ll learn more about this trick and geometric series in Chapter 7. Use this to show that there
are infinitely many even perfect numbers if and only if there are infinitely many Mersenne
primes. Then use your answer from # 3.

• Hint for #10: Remember, this is a * problem. Actually, this is a ** problem!

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