Bijective Functions Modulo N

For any integer $n,$ a function $f:\mathbb{Z} \rightarrow \mathbb{Z}$ is called bijective modulo $n$ if the remainders of $\{f(0), f(1), \cdots , f(n-1) \}$ upon division by $n$ are equivalent to $\{0, 1, \cdots , n-1 \}$ in some order.

A pair of integers $(a,b)$ is called friends with $n$ if the function $f(x) = ax^3 + bx$ is bijective modulo $n.$ Find the number of primes $p < 50$ which satisfy the following property:

• If $(a,b)$ is any pair of integers which is friends with $p,$ it is friends with $n$ for infinitely many integers $n.$

Details and assumptions

• As an explicit example, consider the function $f(x) = 7x$ and take $n=4.$ We have $\{ f(0) , f(1), f(2), f(3) \} = \{0, 7, 14, 21\}.$ Upon division by $4,$ these numbers leave remainders $\{0, 3, 2, 1\}$ respectively, which is equivalent to $\{0, 1, 2, 3\}$ in another order. Thus, $f(x) = 7x$ is bijective modulo $n=4.$

• You may refer to a list of primes .

• This problem is not entirely original.