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Blast into Math! Prime nummers: indestructimle muilding mlocks
5.5 Exercises
1. Show that if p is prime, a ∈ Z , and p|a , then p |a .
n
n
n
2. Show that if a natural number n> 1 is not prime, then there is a natural number d with
√
1 <d ≤ n such that d|n . Use this to prove that to determine whether a natural number
n> 1 is prime, it is sufficient to check whether n is divisible by any of the numbers d
√
with 1 <d ≤ n .
3. * Mersenne primes are those prime numbers which are one less than a power of 2, for
example 7= 2 − 1. What are the first 6 Mersenne prime numbers? Twin primes are a
3
pair of primes that are 2 away from each other, like 5 and 7, or 17 and 19. What are the
first 4 pairs of twin prime numbers? Are there infinitely many Mersenne primes? Are there
infinitely many twin primes?
4. Show that the sum of two consecutive prime numbers bigger than 2 always has at least three
(not necessarily distinct) prime factors.
5. Instead of writing a positive integer as the product of prime numbers, we could write it as
the sum of prime numbers. For example,
5= 2+ 3.
Is it always possible to write x ∈ N as the sum of two (not necessarily distinct) primes, if
x ≥ 4? Why or why not? Prove your answer.
√
6. Prove that 2 /∈ Q .
7. Prove that there are infinitely many correct answers to # 10 from the previous chapter.
8. There are two things which mathematicians find especially beautiful: simplicity and
structure. A perfect number has a special type of structure because its divisors are related to
it not only by division but also by addition.
Definition 5.5.1. An integer x ∈ N is perfect if it is equal to the sum of all its positive divisors
which are less than x . If {s 1 ,s 2 ,...,s n } are all the positive divisors of x which are less than
x , then x is perfect precisely when
s + s + … +s = x
1 2 n
The smallest perfect number is 6. This is because the positive divisors of 6 are 1, 2, 3 and 6,
so the positive divisors that are less than 6 are 1, 2 and 3, and
1+2+ 3= 6.
What is the next perfect number? Can a prime number be perfect?
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