Difference between revisions of "User:Temperal/The Problem Solver's Resource6"
(→Fermat-Euler Identitity) |
(→Euler's Phi Theorem) |
||
Line 43: | Line 43: | ||
===Euler's Phi Theorem=== | ===Euler's Phi Theorem=== | ||
− | If <math> | + | If <math>(a,m)=1</math>, then <math>a^{\phi{m}}\equiv1\pmod{m}</math>, where <math>\phi{m}</math> is the number of relatively prime numbers lower than <math>m</math>. |
===Errata=== | ===Errata=== |
Revision as of 20:38, 10 January 2009
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 6. |
Number Theory
This section covers number theory, specifically Fermat's Little Theorem, Wilson's Theorem, and Euler's Totient Theorem.
Definitions
- if is the remainder when is divided by to give an integral amount. Also, this means b divides (n-a).
- (or divides ) if for some integer .
Special Notation
Occasionally, if two equivalent expressions are both modulated by the same number, the entire equation will be followed by the modulo.
refers to the greatest common factor of and refers to the lowest common multiple of .
Properties
For any number there will be only one congruent number modulo between and .
If and , then .
Fermat's Little Theorem
For a prime and a number such that , . A frequently used result of this is .
Example Problem 1
Find all primes p such that .
Solution
Firstly, p=2 clearly does not work. Now, as all other primes are odd, and hence . After adding one, we have since p divides . However, that means p must divide 3, so the only prime possible is 3. Indeed, is a multiple of 3.
Wilson's Theorem
For a prime , .
Example Problem 2=
Let be an integer such that . Find the remainder when is divided by .
Solution
After multiplying through by , we know that every term on the left-hand-side will be divisible by 13 except for . We wish to find the remainder when is divided by 13. From Wilson's Theorem, we know that so we consider (mod 13). Thus, the remainder is which comes out to be 7. Thus, our answer is 7.
Euler's Phi Theorem
If , then , where is the number of relatively prime numbers lower than .
Errata
An integer n is a quadratic residue (mod m) if and only if there exists an integer p such that . All quadratic residues are or and , , or . All cubic residues (mod 9) are 0, 1, or -1.