# 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.

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