Thread: Difficulties proving that a quadratic expression is composite

I am trying to prove the following statement:
There is a quadratic f(n) = n² + 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.

Originally Posted by matemauch

I am trying to prove the following statement:
There is a quadratic f(n) = n² + bn + c with positive coefficients a and b, such that f(n) is composite.

Where is any a?

I am sorry for the typo. It should be "...with positive integers b and c, such that..."

Thanks !

Originally Posted by Plato

Where is any a?

Sorry for the typo. I think now it is correct.

Take b= 3, c= 2. Then n^2+ 3n+ 2= (n+1)(n+2) is, for any n, the product of two integers and so composite.

Originally Posted by HallsofIvy

Take b= 3, c= 2. Then n^2+ 3n+ 2= (n+1)(n+2) is, for any n, the product of two integers and so composite.

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

Originally Posted by Salahuddin559

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

Assume x and y be integers

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