Number Theory, with a proof discussed in class (not homework)

In summary, Number Theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers. Prime numbers are numbers that are only divisible by 1 and themselves, making them unique and important in number theory. A common proof for prime numbers is the Sieve of Eratosthenes, and the Fundamental Theorem of Arithmetic states that every positive integer can be represented as a unique product of prime numbers. The Goldbach Conjecture, which states that every even integer can be expressed as the sum of two prime numbers, has not been proven but has been tested for large numbers and held true.
  • #1
Worded.Mouse
2
0
Prove that ordda | ordma, when d|m.

Some conditions are 1 ≤ d, 1 ≤ m, and gcd(a,d)=1.

What I have so far:

let x=ordma, which gives us ax[itex]\equiv[/itex] 1 (mod m) [itex]\Rightarrow[/itex] ax=mk+1 for some k[itex]\in[/itex]Z

Let m=m'd. Then ax=mk+1=d(m'k)+1
 
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  • #2
Could someone give me a hint?
 

What is Number Theory?

Number Theory is a branch of mathematics that deals with the properties and relationships of numbers, especially integers.

What are prime numbers?

Prime numbers are numbers that are only divisible by 1 and themselves. In other words, they have no factors other than 1 and itself. Examples of prime numbers include 2, 3, 5, 7, and 11.

How do we prove that a number is prime?

A common proof for prime numbers is the Sieve of Eratosthenes, which involves systematically eliminating all multiples of a number until only the number itself remains. If the number cannot be eliminated, then it is prime.

What is the Fundamental Theorem of Arithmetic?

The Fundamental Theorem of Arithmetic states that every positive integer can be represented as a unique product of prime numbers. This means that every number has a unique prime factorization.

What is the Goldbach Conjecture?

The Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. While this conjecture has not been proven, it has been tested for numbers up to 4 x 10^18 and has held true.

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