# The prime next door

Let us define $$Z(n)$$ , such that for the input of natural $$n$$ , $$Z(n)$$ gives the immediate prime number next to $$n$$ .

Then $a^{Z(n)} \equiv a \mod n$

=> $$a$$ and $$n$$ belong to the set of natural numbers where $$n$$ is not a prime number and greater than one ..

Note by Chinmay Sangawadekar
1 year, 7 months ago

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The claim is not true.

For example, with $$n = 32, Z(n) = 37$$, we have $$3^{37} \equiv 19 \pmod{32}$$.

I believe what happened was that you were testing small cases where the prime factors of $$n$$ were small and distinct, which combined with euler's theorem led to this being true in several cases. As such, I went with a prime power, and then used one that was large enough.

Staff - 1 year, 7 months ago

Awesome result! Is it original?

I'll try proving it.

- 1 year, 7 months ago

Yeah it is , but it is disproved for large values :( .

- 1 year, 7 months ago