MathDB

Version 1. Find all primes

p

satisfying the following conditions:
(i)

\frac{p+1}{2}

is a prime number. (ii) There are at least three distinct positive integers

n

for which

\frac{p^2+n}{p+n^2}

is an integer.
Version 2. Let

p \neq 5

be a prime number such that

\frac{p+1}{2}

is also a prime. Suppose there exist positive integers

a <b

such that

\frac{p^2+a}{p+a^2}

and

\frac{p^2+b}{p+b^2}

are integers. Show that

b=(a-1)^2+1

N6