Consider the following interesting list of integers:
P1 = 43 = 41 + 2,
P2 = 47 = 43 + 4,
P3 = 53 = 47 + 6, … ,
in which Pn is obtained by adding 2n to Pn-1.
It so happens that there is a quadratic function F(x) with the property that
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:
P0 = 41 = 41 + 0 = 41 + 0
P1 = 43 = 41 + 2 = 41 + 12 + 1
P2 = 47 = 41 + 6 = 41 + 22 + 2
P3 = 53 = 41 + 12 = 41 + 32 + 3
P4 = 61 = 41 + 20 = 41 + 42 + 4
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Pn = 41 + n2 + n = n2 + n + 41
Even though the instances we’re looking at are all prime, Pn is not always prime.
If n = 41, Pn = 412 + 41 + 41 = 41(41 + 2) = 41 · 43, which is not prime.