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.

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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.

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Where is any a?

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

Thanks !

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Originally Posted by Plato

Where is any a?

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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.

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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.

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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

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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

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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