The Equivalence of Wilson's Theorem and sin(πΩ) = 0

Key Points

• To establish a new criterion for determining if an integer is prime through Wilson's theorem and the sine function.
• Developed a criterion using the equation sin(π*Ω) = 0, where Ω = [(p-1)! + 1]/p.
• Examined the relationship between factorials and prime numbers through trigonometric functions.
• A necessary and sufficient condition for primality is established, showing sin(π*Ω) = 0 for prime integers.
• This criterion effectively links elementary number theory to the behavior of sine functions.