Question 186
Consider a sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfying for each $n \ge 3$ the condition
$$
a_n = a_1a_2 + a_2a_3 + \dots + a_{n-2}a_{n-1} - 1.
$$
a) Prove that some prime number is a divisor of infinitely many terms of this sequence.
(Tomáš Bárta)
b) Prove that there are infinitely many such prime numbers.
$$
a_n = a_1a_2 + a_2a_3 + \dots + a_{n-2}a_{n-1} - 1.
$$
a) Prove that some prime number is a divisor of infinitely many terms of this sequence.
(Tomáš Bárta)
b) Prove that there are infinitely many such prime numbers.