Disprove the following statements:
(a) If $p$ is prime then $p + 2$ is prime.
(b) The integer $n^2 + 1$ is prime for $n = 2m$.
(c) The integer $n^2 − 1$ is composite.
(d) The quadratic $4n^2 − 2n + 1$ where $n$ is a natural number produces primes.
(e) The Euclid number $N$ given by $N = (2 × 3 × 5 × 7 × ⋯ × P) + 1$ is prime where $P$ is a prime number.
We disprove the statements using counter-examples.
(a) $p=7$ is prime, but $p + 2 = 7+2 = 9$ is not prime.
(b) $m=4$ gives $n=2m=8$. Here $n^2+1 = 8^2 +1 = 65$ which is not prime.
(c) $n=2$ gives $n^2-1 = 2^2 - 1 = 3$ which is prime, not composite.
(d) $n=4$ gives $4n^2 -2n +1 = 64-8+1 = 57 = 19*3$, so is not prime.
(e) Consider $P=13$. Then $N = (2 \times 3 \times 5 \times 7 \times 11 \times 13) + 1 = 30031 = 59 \times 509$, so is not prime.