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I'm a programmer, a newbie on math. I'm trying to code to list twin prime.

I've found this: $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$

The pair (m, m + 2) is twin prime, iff 4((m − 1)! + 1) ≡ −m (mod m(m + 2)).

However, if I set m=5 and then, 

4((m - 1)! + 1) = 4 * (4! + 1) = 4 * 25 = 100
-m (mod m(m + 2)) = -5 mod 35 = 35 - (5 mod 35) = 30

Well, 100 does not equal to 30, right? But (5, 7) is a twin prime.

Did I do something wrong?

Bill Dubuque
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Yves
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1 Answers1

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You didn't do anything wrong. However, note that $30 \equiv 100 \pmod{35}$ (in general, $a \equiv b \pmod c$ means $c$ divides evenly into $a - b$, so $35$ divides $100 - 30 = 70$ in this case). Thus, the $2$ values are effectively equivalent, at least in terms of modulo $35$.

John Omielan
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