MathDB

Let

c \ge 1

be an integer. Define a sequence of positive integers by

a_1 = c

and

a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c

for all

n\ge 1

. Prove that for each integer

n \ge 2

there exists a prime number

p

dividing

a_n

but none of the numbers

a_1 , \ldots , a_{n -1}

.
Proposed by Austria

N7