Question 27
Determine the sets $A$ of positive natural numbers, with at least four elements $a_1 \lt a_2 \lt a_3 \lt \dots$, such that:
$$
\bullet \ a_1 = 1, a_2 = 2;
$$
$$
\bullet \ a_i a_{i+3} + 1 = a_{i+1} a_{i+2}, \text{ for every positive integer } i;
$$
$$
\bullet \ A \text{ contains exactly four prime numbers.}
$$
$$
\bullet \ a_1 = 1, a_2 = 2;
$$
$$
\bullet \ a_i a_{i+3} + 1 = a_{i+1} a_{i+2}, \text{ for every positive integer } i;
$$
$$
\bullet \ A \text{ contains exactly four prime numbers.}
$$