##(p^k -1) \equiv X \mbox{(mod p)}## via Wilson's theorem

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Thread starter DaTario
Start date Jan 18, 2021
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In summary, Wilson's theorem states that for any prime number p, (p-1)! ≡ -1 (mod p). This can be used to simplify equations like (p^k -1) ≡ X (mod p) to -1 ≡ X (mod p), allowing for easier solution of such equations. However, Wilson's theorem only applies to prime numbers and there are other theorems and methods, such as Fermat's little theorem and Euler's theorem, that can also be used to solve similar equations. Wilson's theorem has many applications in mathematics and science, particularly in number theory and modular arithmetic, and has been used to prove other important theorems.

Jan 18, 2021

DaTario

TL;DR Summary: Hi, is there a way to obtain ##(p^k-1)! \equiv X \mbox{ (mod p)}## for ##X## using Wilson's theorem: ##[ (p-1)! \equiv -1 \mbox{(mod p)} ] ##?

Hi All, being ##p## a prime number, is there a way to solve the congruence ##(p^k-1)! \equiv X \mbox{ (mod p)}## for ##X## using Wilson's theorem: $$ (p-1)! \equiv -1 \mbox{(mod p)} $$?

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Jan 18, 2021

fresh_42

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##k=1## is Wilson and ##k>1## means ##(p^k-1)!\equiv 0 \mod p##.

Jan 18, 2021

DaTario

Thanks!

1. What is Wilson's theorem?

Wilson's theorem is a mathematical theorem that states that for any prime number p, (p-1)! is congruent to -1 modulo p. This means that when (p-1)! is divided by p, the remainder will always be -1.

2. How is Wilson's theorem related to (p^k - 1) ≡ X (mod p)?

Wilson's theorem can be used to simplify the expression (p^k - 1) ≡ X (mod p). By applying Wilson's theorem, we can rewrite the expression as (p-1)! ≡ X (mod p). This makes it easier to solve for X, as we only need to find the remainder of (p-1)! when divided by p.

3. What is the significance of (p^k - 1) ≡ X (mod p) via Wilson's theorem?

This expression has several applications in number theory and cryptography. It can be used to test whether a number is a prime or not, and also to generate prime numbers. In cryptography, it is used in the RSA algorithm for encryption and decryption.

4. Can Wilson's theorem be applied to non-prime numbers?

No, Wilson's theorem only applies to prime numbers. If applied to a non-prime number, the result may not be accurate.

5. How can Wilson's theorem be proven?

There are several ways to prove Wilson's theorem, including using mathematical induction, group theory, and combinatorics. One of the simplest proofs involves using the fact that every number has a unique inverse modulo p, and rearranging the terms in (p-1)! to show that it is congruent to -1 modulo p.