Taha’s Detector of Prime Number

This would not qualify as research in any modern academic setting since the result you are claiming is already well-known, and this proof is not even fully correct; moreover, it is very poorly written. It looks like you are attempting to prove what is known as Wilson's Theorem, a classical result in number theory that is often taught at the undergraduate level. It states that p is prime if and only if (p-1)! ≡ -1 (mod p), which is actually slightly stronger than what you have (you only test if (p-1)! is not 0 (mod p)). In your paper, case i) isn't quite correct because if (g-1)!/g is not an integer, you need to actually prove that g cannot be composite. What if g happens to be composite and some prime appears more frequently in g than it does in (g-1)!? For example, if g=4, then (4-1)!/4 is not an integer, but 4 is composite. The result is true if g>4, but you have to actually show that. For case ii), it is trivial if g = ab for some integers 2 <= a,b < g. But what if g is e.g., the square of a prime? Lastly, while this gives a (well-known) primality test, it is extremely inefficient in practice. Suppose you had to test if 1009 was prime. Then this method would involve computing 1008! modulo 1009, and 1008! contains almost 2600 digits. This is way slower in practice than even the standard school method.

Professional & Academic I’m excited to share that my research “Taha’s Detector of Prime Number” has been officially approved and published on Cambridge Open Engage, a platform by Cambridge University Press for early and open research. This is a meaningful milestone for me, and I’m grateful to contribute to the global research community. Looking forward to expanding this work and continuing to explore new mathematical ideas.