A15759. Many Perfect Squares

You are given a set $a_1, a_2, \ldots, a_n$ of distinct positive integers.

We define the squareness of an integer $x$ as the number of perfect squares among the numbers $a_1 + x, a_2 + x, \ldots, a_n + x$ .

Find the maximum squareness among all integers $x$ between $0$ and $10^{18}$ , inclusive.

Perfect squares are integers of the form $t^2$ , where $t$ is a non-negative integer. The smallest perfect squares are $0, 1, 4, 9, 16, \ldots$ .

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 50$ ). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1 \le n \le 50$ ) — the size of the set.

The second line contains $n$ distinct integers $a_1, a_2, \ldots, a_n$ in increasing order ( $1 \le a_1 < a_2 < \ldots < a_n \le 10^9$ ) — the set itself.

It is guaranteed that the sum of $n$ over all test cases does not exceed $50$ .

For each test case, print a single integer — the largest possible number of perfect squares among $a_1 + x, a_2 + x, \ldots, a_n + x$ , for some $0 \le x \le 10^{18}$ .

In the first test case, for $x = 0$ the set contains two perfect squares: $1$ and $4$ . It is impossible to obtain more than two perfect squares.

In the second test case, for $x = 3$ the set looks like $4, 9, 16, 25, 100$ , that is, all its elements are perfect squares.