I am trying to prove the following statement:
There is a quadratic f(n) = n^{2} + bn + c with positive coefficients b and c, such that f(n) is composite.
I am facing difficulties on how to approach this proof and right now I have no idea on how to start. Could you please give me a hint?
Thank you very much in advance.
Thank you very much. I completely forgot about the factorization. Based on your suggestion I elaborated the next line of thought:
Let q and p positive integers.
Define b = x+y, and b is a positive integer
Define c = xy, and c is a positive integer
Then n^2 + bn + c = n^2 + (x+y)n + xy = (n+x)(n+y) is composite.
What do you think of it?
Thank you for your help and your comment.
What are you trying to prove here? I am lost in this, from what you are saying, x and y need not even be rational numbers, ok? Is that fine, having some irrational numbers in the factorization?
Salahuddin
Maths online
It is not always possible that x and y are integers. x and y are such that -x and -y are roots of this quadratic. And those are not always real numbers, even.
Salahuddin
Maths online