MATH 2710, SPRING 2014
HOMEWORK #4
JOHANNA FRANKLIN
This assignment will be due on Thursday, February 27 at the beginning of class. Remember to
show your reasoning and name the classmates you worked with. Your solutions should be detailed
enough that any of your classmates could understand them simply by reading them.
(1) (Chapter 2, #92, rephrased) In this problem, we’re going to think about primality in a
different context. Consider the set of all even integers 2Z = {2n | n ∈ Z}. Just like Z,
2Z is closed under addition, subtraction, and multiplication, but not division (because, for
instance, you don’t get an even integer when you divide 12 by 4). We say that a | b in 2Z
if a and b are both in 2Z and there is a k ∈ 2Z such that ak = b (for instance, 6 | 12 in
2Z because 6 · 2 = 12, but as we saw above, 4 doesn’t divide 12 in 2Z because you can’t
multiply 4 by an even integer and get 12). A prime in 2Z is therefore a positive even integer
that cannot be factored into two even integers.
(a) Find all the primes in 2Z.
(b) Can every positive element of 2Z be expressed as a product of primes in 2Z? Why or
why not?
(c) If a number in 2Z can be factored into primes in 2Z, is that factorization unique?
(2) (Chapter 2, #93, rephrased) Prove that the sum of any two twin primes (that is, two primes
that differ by 2 like 3 and 5 or 11 and 13) has at least 3 prime factors. (These factors do
not have to be different!)
(3) (Chapter 2, based on #106)
(a) Find two consecutive primes that differ by at least 8.
(b) Prove that there are arbitrarily big gaps between consecutive primes. (Hints: To
do this, show that for every n ∈ P, you can find a sequence of consecutive positive
integers of length at least n with no primes. Also keep in mind that the factorial
function (defined on p. 91) could be useful.)
(4) Consider the following statement:
Let p, a, b ∈ Z. Suppose that p is prime, p | a, and p | (a2 + b2 ). Then p | b.
If the statement is true, prove it, and if it is false, provide a counterexample.
(5) The claim below is true, and the proof is mostly correct, but it has been written poorly.
Rewrite this proof to make it clearer and more correct.
Claim. Let a, b ∈ Z. If a | b, then a2 | b2 .
1
2
FRANKLIN
“Proof ”. Since a | b, there is k such that ak = b. This says (ak)(ak) = bb, so we can get
b2
= ` so a2 | b2 .
a2
Suggested problems: Chapter 2: 66, 70-71, 94, 96, 100-101, 105, 108