Mathematics 400c
Homework (due Feb. 4)
A. Hulpke
7) Find a sequence of 99 consecutive integers which are not prime.
8) Compute (without using a Gcd function on a computer) the Gcd of a = 67458 and
b = 43521 and express it in the for xa + yb.
n
9) The n-th Fermat number is defined to be Fn = 22 + 1. Show that for n 6= m the Gcd
(Fn , Fm ) = 1. Conclude from this that there are infinitely many primes.
10) a) Determine all positive integers that have exactly 8 divisors (including 1 and itself).
b) Determine all positive integers, whose divisors sum up to 60. (In the notation of problem
5: All n such that σ(n) = 60.)
11) Prove that if a ≡ b (mod m) and c ≡ d
a) a + c ≡ b + d (mod m),
b) a · c ≡ b · d (mod m).
(mod m) then
12∗ ) (GAP) We want to check how many pairs (x, y) with 1 ≤ x, y ≤ n have Gcd One. Call
the number of these
V (n) = |{(x, y) | 1 ≤ x, y ≤ n, gcd(x, y) = 1}|
In GAP one can check this for example by the commands (for n = 20):
gap> n:=20;
20
gap> pairs:=Cartesian([1..n],[1..n]);;
gap> Vn:=Number(pairs,i->Gcd(i[1],i[2])=1);
255
Check the ratio
V (n)
n2
for increasing n. Does this ratio have a limit if n → ∞?
Problems marked with a ∗ are bonus problems for extra credit.