1Prove that there exists a polynomial of the form fnn bnc wh

1)Prove that there exists a polynomial of the form f(n)=n² +bn+c, where b and c are positive integers, such that f(n) is composite

2) Show that for every positive integer a there exists a positive integer b such that ab+1 is a perfect square. Let Z+ be the set of all positive integers and a, b Z+

Solution

1) Proof :-

Let f(x) be a non-constant polynomial with integer coefficients.
Without loss of generality we can assume that its lead coefficient is positive.

It is not hard to show that there is a positive integer N such that for all nN, we have f(n)>1,
and such that f(x)is increasing for xN. (For large enough x, the derivative f(x) is positive.) = 2x+b

Let f(N)=q. Then f(N+q) is divisible by q.
But since f(x) is increasing in [N,), we have f(N+q)>q.
Thus f(N+q) is divisible by qq and greater than q, so must be composite.

Remark: One can remove the \"size\" part of the argument.
For any b, the polynomial equation f(x)=b has at most d solutions,
where d is the degree of f(x). So for almost all integers nn, f(n) is not equal to 0, 11, or 1.

Let N be a positive integer such that f(n) is different from 0, 11, or 1 for all nN.
Let f(N)=q.
Consider the numbers f(N+kq), where k ranges over the non-negative integers.
All the f(N+kq) are divisible by q.
But since the equations f(n)=±q have only finitely many solutions,
there is a k (indeed there are infinitely many kk) such that f(N+kq) is not equal to ±q,
but divisible by q. Such a f(N+kq) cannot be prime.