Problem 1

For each integer

a_0>1

, define the sequence

a_0,a_1,a_2,\ldots

for

n\ge 0

a_{n+1}=\begin{cases}\sqrt{a_n} & \text{if }\sqrt{a_n}\text{ is an integer},\\ a_n+3 & \text{otherwise}.\end{cases}

Determine all values of

a_0

for which there exists a number

A

such that

a_n=A

for infinitely many values of

n

Step 2 of 5: Divisibility by 3 is an invariant

Previous step Next step

3\mid a_n \iff 3\mid a_0

Detailed analysis

Adding $3$ never changes the residue modulo $3$ . If $a_n$ is a perfect square, write $a_n=k^2$ ; since $3$ is prime, $3\mid k^2$ exactly when $3\mid k$ , so taking the square root neither creates nor destroys a factor of $3$ . Hence whether $3\mid a_n$ is the same for every $n$ as it is for $a_0$ .