Download 24. Prove that every integer greater than 11 can be expressed as th

Math 320 - Dr. Miller - Solutions to HW #4 - 2/1/08
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24. Prove that every integer greater than 11 can be expressed as the sum of two composite
numbers.
Proof: Let n ∈ Z be greater than 11. We’ll consider two cases.
n is even. Then n ≥ 12 > 8. Indeed, n − 8 is a positive, even integer greater than or
equal to 4, and all such integers are composite. Then n = 8 + (n − 8) expresses n
as the sum of the composite numbers 8 and n − 8.
n is odd. Then n ≥ 13 > 9. In this case, n − 9 is a positive, even integer greater than
or equal to 4, and n = 9 + (n − 9) expresses n as the sum of the composite numbers
9 and n − 9.
26. Prove or disprove the following statements.
(a) If p is a prime number, then 2p − 1 is a prime number.
Counterexample: 11 is prime, but 211 − 1 = 2047 = 23 · 89 is composite.
(b) If 2p − 1 is a prime number, then p is a prime number.
Proof: (By contrapositive) Suppose p is composite with p = ab for some integers
a and b strictly between 1 and p. Note that the number 2ab − 1 is positive (ab =
p > 1 because p is composite, whence 2ab − 1 ≥ 2 − 1 = 1) and certainly an
integer. Reference to algebra texts or experimentation shows that 2ab − 1 factors as
(2a − 1)(2(b−1)a + 2(b−2)a + · · · + 22a + 2a + 1) and by closure both are integers. Now
since a > 1, the factor 2a − 1 does not equal 1, and since b > 1, the other factor has
at least two terms; that is, it is equal to the final 1 plus at least one more number.
Thus, the second factor does not equal 1 either. Both factors are positive – the first
because a > 1 implies 2a ≥ 2 and the second because it is a sum of strictly positive
numbers. Thus 2ab − 1 has positive integer factors other than 1 and itself and must
therefore be composite.