Generalising Prime numbers

Number Theory Level 2

True or false :

$\ P(n) = n^2 + n + 41$ is a prime for all integers $n$ where $0 \leq n \leq 100$ .

False True

3 solutions

Brian Charlesworth
Dec 27, 2015

$P(n)$ is prime for all integers $n$ such that $0 \le n \le 39$ , but

$P(40) = 40^{2} + 40 + 41 = 40*(40 + 1) + 41 = (40 + 1)*41 = 41^{2}$ is composite.

Can you prove that it's an prime for natural numbers less than 40?

Arulx Z - 5 years, 5 months ago

It was Euler that first noticed that $n^{2} + n + 41$ was a prime generating polynomial . This is a surprisingly complex topic, as evidenced by this paper .

Brian Charlesworth - 5 years, 5 months ago

Thanks for reply. Links are very helpful (and pretty complex too).

Arulx Z - 5 years, 5 months ago

Ankit Nigam
Dec 29, 2015

In a simple way just put $n = 41$ , we get $41^{2} + 41 + 41$ and obviously $41$ is a factor itself. Thus its not a prime for $0$ $\leq$ $n$ $\leq$ $100$

Nagarjuna Reddy
Dec 27, 2015

n (n+1)+41 iif we rewrite the given..n(n+1) is always even.41 +even always not a prime number.

Generalising Prime numbers

3 solutions

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