Wilson’s Theorem

Wilson's Theorem is a classical theorem in Number Theory. It has also an interesting fact associated with it that the theorem is stated by a teacher and student pair, i.e., Ibn al-Hayatam and John Wilson, respectively. However, it was proved later on by Lagrange. Wilson’s Theorem is of great importance in Number Theory. The applications of Wilson’s Theorem, its limitations, and Wilson’s Theorem examples will be discussed here in detail for a clear understanding of the theorem. So, let us discuss the theorem.

History of John Wilson

John Wilson

• Name: John Wilson

• Born: 6 August 1741

• Died: 18 October 1793

• Field: Mathematics

• Nationality: British

Wilson's Theorem Statement

Wilson Theorem states that if $p$ is any natural number greater than 1, then $p$ is said to be a prime number if and only if the product of all the positive integers less than $p$ is one less than a multiple of number $p$.

Mathematically, according to Wilson's Theorem, let $p>1$. The number $p$ is prime if and only if

$(p-1) !=-1(\bmod p)$

or,

$(p-1) !=(p-1)(\bmod p)$

Wilson's Theorem Proof

A natural number p > 1 is a prime number if and only if the product of all the positive integers less than p is one less than a multiple of p.

Suppose $p$ is prime. If $k \in\{1, \ldots, p-1\}$, then $k$ is relatively prime to $p$. So, there are integers $a$ and $b$ in the way that

$\Rightarrow a k+b p=1, \text { or } a k=1(\bmod p) \text {. }$

Reducing $a \bmod p$,

We may assume $a \in\{1, \ldots, p-1\}$. Thus, every element of $\{1, \ldots, p-1\}$ has a reciprocal $\bmod p$ in this set. We know that only 1 and $p-1$ are their reciprocals. Thus, the elements $2, \ldots, p-2$ must pair up into pairs $\left\{x, x^{-1}\right\}$. It is obvious that their product is 1.

Hence,

\[\Rightarrow 1 \cdot(p-1)=p-1 \]

\[ \Rightarrow-1(\bmod p)\]

$\Rightarrow(p-1) !=1 \cdot 2 \cdots(p-2) \cdot(p-1)$

Now suppose $(p-1) !=-1(\bmod p)$

We want to show that $p$ is prime.

Begin by rewriting the equation as $(p-1) !+1=k p$.

Suppose $p=a b$. We may take $1 \leq a, b \leq p$. If $a=p$, then the factorization is trivial, so suppose $a<p$. Then, $a \mid(p-1)$ ! (since it's one of $\{1, \ldots, p-1\})$ and $a \mid p$,

So,

\[\Rightarrow(p-1) !+1=k p\] shows \[a \mid 1.\]

Therefore, $a=1$.

Limitations of Wilson's Theorem

Wilson's theorem is not applicable in the case of composite number 4 because by applying Wilson's Theorem in the case of 4, we get the remainder 2 instead of 1.

Applications of Wilson's Theorem

Wilson’s Theorem has a wide range of applications in the changing world. As the world is shifting to digitization, security is the key concern.

• Wilson’s Theorem is used in cryptography for coding-decoding.

• Wilson's Theorem is applicable in the case of both prime and composite numbers. Hence, widens the base of its applications.

Wilson’s Theorem Examples

1. What will be the remainder when 568! is divided by 569?

Ans: According to Wilson's theorem, we have,

For prime number ' $p$ ', we have

$\Rightarrow(p-1) ! \bmod p=(p-1)$

In this case, 569 is a prime number.

Thus,

$\Rightarrow 568 ! \bmod 569=568 \text {. }$

Hence, when 568 ! is divided by 569,

$\Rightarrow$ we get 568 as the remainder.

2. What will be the remainder when 225! is divided by 227?

Ans: We know that for prime number ' $p$ ', we have

$\Rightarrow(p-2) ! \bmod p=1 \text {. }$

In this case, 227 is a prime number.

Thus, $225 ! \bmod 227$ will be equal to 1.

In other words,

when 225! Is divided by 227,

$\Rightarrow$ we get the remainder as 1.

Important Formulas to Remember

• For $p>1$ and $p$ is prime number: $(p-1) !=-1(\bmod p)$

Important Points to Remember

• Wilson’s Theorem is applicable only in the case of a prime number.

• A number is said to be prime if it is divisible by 1 and itself only.

Conclusion

In the article, we have discussed the proof of Classical Wilson's Theorem in detail and its applications in the real world. Wilson’s Theorem forms a fundamental tool of Number Theory. It reduces our calculation work, which would even have taken hours without this theorem. Finally, we can say that Wilson's Theorem is a gem of Mathematics.