Difficulties proving that a quadratic expression is composite

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.

Re: Difficulties proving that a quadratic expression is composite

Quote:

Originally Posted by

**matemauch** I am trying to prove the following statement:

There is a quadratic f(n) = n^{2} + bn + c with positive coefficients a and b, such that f(n) is composite.

Where is any a?

Re: Difficulties proving that a quadratic expression is composite

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

Thanks !

Re: Difficulties proving that a quadratic expression is composite

Quote:

Originally Posted by

**Plato** Where is any a?

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

Re: Difficulties proving that a quadratic expression is composite

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.

Re: Difficulties proving that a quadratic expression is composite

Quote:

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.

Re: Difficulties proving that a quadratic expression is composite

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

Re: Difficulties proving that a quadratic expression is composite

Quote:

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

Re: Difficulties proving that a quadratic expression is composite

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