For 2), assume p,q are twin primes, that is .
Then, by the binomial theorem:
Now, since for each , we get that
Can you finish from here?
Problem 15
A pair (x,y) of positive integers is called square if x + y and xy are both perfect squares. For example, pair (5,20) is square since 5 + 20 = 52 and 5 x 20 = 102. Prove that no square pairs exists in which one of its numbers is 3.
Problem 16
Primes p and q are called twin primes, if q = p + 2. Prove that the numbers p^4 + 4 and q^4 + 4 are never relatively prime, if p and q are twin primes.
Hehe, I like that term ... morning brainfart
aman_cc, how did you get that 3|b^2 => 9|b^2?
I got that also, but I'm just wondering if we did it the same way (I like how alot (in my limited experience) of things in number theory can be proven in different ways )