Consider the following interesting list of integers:

*P*

_{0}= 41,

*P*

_{1}= 43 = 41 + 2,

*P*

_{2}= 47 = 43 + 4,

*P*

_{3}= 53 = 47 + 6, … ,

in which *P*_{n} is obtained by adding 2*n* to *P*_{n-1}.

It so happens that there is a quadratic function *F*(*x*) with the property that *P*_{n} = *F*(*n*)*n*. Find a formula for *F*(*x*).

Once you have the formula, consider the question of whether or not *F*(*n*) is a prime number for every nonnegative value of *n*.

The trick is to rewrite each integer in terms of the first value (41) and look for a pattern that resembles a quadratic function:

P_{0} = 41 = 41 + 0 = 41 + 0

P_{1} = 43 = 41 + 2 = 41 + 1^{2} + 1

P_{2} = 47 = 41 + 6 = 41 + 2^{2} + 2

P_{3} = 53 = 41 + 12 = 41 + 3^{2} + 3

P_{4} = 61 = 41 + 20 = 41 + 4^{2} + 4

.

.

.

P_{n} = 41 + *n*^{2} + *n* = *n*^{2} + *n* + 41

Even though the instances we’re looking at are all prime, P_{n} is not always prime.

If *n* = 41, P_{n} = 41^{2} + 41 + 41 = 41(41 + 2) = 41 · 43, which is not prime.