Wilson's Theorem states that a positive integer $p > 1$ is prime if and only if $(p-1)! \equiv -1 \pmod p$, showing a relationship between factorials and prime numbers. Finding it curious, today I started trying to understand why it happens by looking for other relationships between them; I ended up dividing the factorial by its analogous sum function, that generates the triangular numbers (the $nth$ triangular number is the sum of all positive integers up to $n$, inclusive), and found out that they are related indeed.

Let $T_n$ denote the $nth$ triangular number and $q=p-1$. I noticed that the following congruences $$A:\ q!\equiv T_q-1\pmod{T_p},$$ $$B:\ q!\equiv q \pmod{T_q},$$ and $$C: (q-1)!\equiv T_{(q-1)}\pmod{T_q}$$ hold, apparently, if and only if $p>2$ and $p$ is prime.

For all other cases, i.e. when $p$ isn't prime, $T_p\mid q!$, $T_q\mid q!$, and $T_q\mid (q-1)!$. I found the last congruence (congruence $C$) especially interesting, because it means that the primality of a positive integer $p>2$ can be proven without making any calculation involving $p$ itself. If true, it means: let $q$ be an even integer such that $q>2$, then $q+1$ is prime if and only if $T_q\nmid (q-1)!$.

I can't prove or disprove these congruences, however. I couldn't find anything related to them on internet either. Can anyone help? Also, is there any known primality test that doesn't involve directly the integer being tested?


  • $\begingroup$ Please not the proper use of \pmod, as in my edit to the question. $\qquad$ $\endgroup$ – Michael Hardy May 21 '16 at 23:11
  • $\begingroup$ Upto which prime did you verify the conjecture ? $\endgroup$ – Peter May 23 '16 at 19:18
  • $\begingroup$ I ran a script and found a counterexample to apparently all three congruences at $p=23$. $\endgroup$ – Cardioid_Ass_22 Mar 3 at 15:30

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