Find all the numbers N,P that satisfy the following conditions

Question

Find all the numbers $N$, $P$ that satisfy:

• $N$ is a whole number
• $P$ is a prime number
• $N = \frac{2P^2 -2P}{P+1}$

Answer 1

First solution: $ N = \frac{2P^2-2P} { P+1} = 2P - 4 + \frac{4}{p+1}$.

Hence for $N$ to be an integer, $ \frac{ 4}{P+1}$ is an integer, which means that $P +1 = 1, 2, 4$. The only prime is $P = 3$, which gives $N=3$.

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Second solution: Deal with $P = 2$ seperately, $N = \frac{4}{3}$ is not an integer, so no solution.

Hint: $\gcd(P, P+1) = 1$.

Hint: $\gcd (P-1, P+1) = 2$ for $P \geq 3$.

Since $P+1$ shares no other common factor with $P^2 - P$ other than possibly 2, this means that $\frac{P+1}{2}$ must be a factor of the only remaining term in the numerator, which is 2.

Hence $P+1$ is a factor of 4, which again gives us the only solution $P=3$.