Anonymous user
Carmichael 3 strong pseudoprimes: Difference between revisions
m
added whitespace before the TOC (table of contents), added a ;Task: (bold) header, added other whitespace to the task's preamble.
No edit summary |
m (added whitespace before the TOC (table of contents), added a ;Task: (bold) header, added other whitespace to the task's preamble.) |
||
Line 1:
{{task}}
A lot of composite numbers can be separated from primes by Fermat's Little Theorem, but there are some that completely confound it.
The [[Miller-Rabin primality test|Miller Rabin Test]] uses a combination of Fermat's Little Theorem and Chinese Division Theorem to overcome this.
The purpose of this task is to investigate such numbers using a method based on [[wp:Carmichael number|Carmichael numbers]], as suggested in [http://www.maths.lancs.ac.uk/~jameson/carfind.pdf Notes by G.J.O Jameson March 2010].▼
Task:
▲The purpose of this task is to investigate such numbers using a method based on [[wp:Carmichael number|Carmichael numbers]], as suggested in [http://www.maths.lancs.ac.uk/~jameson/carfind.pdf Notes by G.J.O Jameson March 2010].
Find Carmichael numbers of the form:
:::: <math> Prime_1 \times Prime_2 \times Prime_3 </math>
<br>(See page 7 of [http://www.maths.lancs.ac.uk/~jameson/carfind.pdf Notes by G.J.O Jameson March 2010] for solutions.)
For a given <math> Prime_1 </math>
<tt>for 1 < h3 < Prime<sub>1</sub></tt>
: <tt>for 0 < d < h3+Prime<sub>1</sub></tt>
:: <tt>if (h3+Prime<sub>1</sub>)*(Prime<sub>1</sub>-1) mod d == 0 and -Prime<sub>1</sub> squared mod h3 == d mod h3</tt>
:: <tt>then</tt>
::: <tt>Prime<sub>2</sub> = 1 + ((Prime<sub>1</sub>-1) * (h3+Prime<sub>1</sub>)/d)</tt>
::: <tt>next d if Prime<sub>2</sub> is not prime</tt>
::: <tt>Prime<sub>3</sub> = 1 + (Prime<sub>1</sub>*Prime<sub>2</sub>/h3)</tt>
::: <tt>next d if Prime<sub>3</sub> is not prime</tt>
::: <tt>next d if (Prime<sub>2</sub>*Prime<sub>3</sub>) mod (Prime<sub>1</sub>-1) not equal 1</tt>
::: <tt>Prime<sub>1</sub> * Prime<sub>2</sub> * Prime<sub>3</sub> is a Carmichael Number</tt>
<br><br>
|