What is the Proof for (p-1)!≡±1 (mod p) in Number Theory?

In summary, number theory is a branch of mathematics that studies patterns and relationships within numbers. Proofs are essential in number theory as they provide rigorous evidence for the validity of mathematical statements. They often rely on specific number properties and techniques and can be more abstract compared to other branches of mathematics. Some famous proofs in number theory include Euclid's proof of the infinitude of primes, Fermat's Last Theorem, and Pythagoras' proof of the irrationality of the square root of 2. To improve skills in proving theorems in number theory, a strong foundation in various mathematical concepts and practice with proof techniques is necessary.
  • #1
koukou
5
0
Recall the definition of n! (read n factorial"):
n! = (n)(n-1)(n-2) ….(2)(1) =∏(k)
In both (a) and (b) below, suppose p≥3 is prime.
(a) Prove that if x∈ Zpx is a solution to x square ≡1 (mod p), then x ≡±1 (mod p).
(b) Prove that (p-1)!≡±1 (mod p)

Zpx x shoud be above p

a and b looks like some theorem proof
 
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  • #2
Hints:
1. factorise x^2-1=(x-1)(x+1).

2. this theorem is called wilson theorem.
 
  • #3
Hint: (2) can be looked at as a case of a and its inverse. The first part, (1) plays a special role in that.
 

1. What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It involves studying patterns and structures within numbers and their interactions with each other.

2. What is the importance of proofs in number theory?

Proofs are essential in number theory as they provide rigorous and logical evidence for the validity of mathematical statements. They help to establish the truth of conjectures and provide a deeper understanding of the underlying concepts and patterns in number theory.

3. How are proofs in number theory different from other branches of mathematics?

Proofs in number theory often rely on specific number properties and techniques, such as prime factorization, modular arithmetic, and the fundamental theorem of arithmetic. They also tend to involve more abstract concepts and can be more difficult to visualize compared to other branches of mathematics.

4. What are some famous proofs in number theory?

Some famous proofs in number theory include Euclid's proof of the infinitude of primes, Fermat's Last Theorem, and the proof of the irrationality of the square root of 2 by Pythagoras. Other notable proofs include the Goldbach Conjecture, the Twin Prime Conjecture, and the Four Color Theorem.

5. How can one improve their skills in proving theorems in number theory?

Improving skills in number theory proofs requires a strong foundation in algebra, geometry, logic, and other mathematical concepts. Practice and exposure to a variety of proof techniques, such as induction, contradiction, and direct proof, can also help to improve skills in proving theorems in number theory.

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