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Old 2003-12-08, 08:33   #1
Evgeny Dolgov
 

16C16 Posts
Default Effective way to generate prime numbers (infinitive)

Hi all!


why i'm choosing this form of numbers

10^(n*2)+10^(n)+3

1. simple view

2. symmetric and asymmetric in one time
well balanced

3. Predictable step (distribution)

4. Using this tool http://www.alpertron.com.ar/ECM.HTM

i have found that if the test number is compiste
(not prime) it takes little time to fail this number

if the number is fail (not prime) it breaks very fast on parts

but if this tool show is unknown and
it takes more than 2-3 seconds on numbers with in
685 digits - i know 100% that number is prime!!!
and tests show this. (i fully tested)

now i'm begining to test numbers with 9555-10000 digits
now fail test takes about 1-2 minute - in most hard
but in simple - 5 seconds for fail - but if test takes
nore than 4-5 (10) minutes with 100% guarantee
i say that this number is PRIME

(before all this my try was the numbers like 11119 1113 10003 ... - it's wrong numbers they are not productive like
that 10....010....03 form or in exponenta 10^(2*n)+10^(n)+3 )

SO MAIN FEATURE - IF TEST NUMBERS in
(10....010....03 form or in exponenta 10^(2*n)+10^(n)+3 )
ARE NOT PRIME
THEY FAIL VERY FAST!!!!!

so the effective search:

use test algorithm like in tool http://www.alpertron.com.ar/ECM.HTM
and comparing relatively time of success test and time of fail test

example if test number - "685 digits" - and test is not failing after 10 seconds
the number is PRIME. (100%)
(main feature)

Evgeny Dolgov

You can test this

100000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000100000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000
0000000000003

this number is prime - the tool (http://www.alpertron.com.ar/ECM.HTM) is showing
"is unknown" - 2 hours and then - is PRIME!!!

if you add or remove one digit "0" at the end of this number
it fails very fast - not more than 10 seconds
on pentium 4 computer.


So repeat
if test (10^(2*n)+10^(n)+3 form number) using algorithm (http://www.alpertron.com.ar/ECM.HTM)
is not failing
after relatively short time
the number is PRIME. (100%)
(main feature)

Evgeny Dolgov

Last fiddled with by Evgeny Dolgov on 2003-12-08 at 08:35
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Old 2003-12-08, 09:25   #2
TauCeti
 
TauCeti's Avatar
 
Mar 2003
Braunschweig, Germany

2×113 Posts
Default

You algorithm produces numbers of a special form. Some of them are prime and look interesting in their decimal expansion.

By the way: There are a lot of numbers, that look 'interesting' and are prime. Take for example the number 10000...(repeat the '0' for 39017 times)...00001. It is prime. More examples at Prime curious

The problem with your algorithm is, that not every number it produces is prime. So you have to check with a 'prime-checker', if it is prime or not.

For random numbers of the lenght of 10 million decimals, there is no known algorithm that decides this question in a practicable timeframe. Only for very special prime-candidates (Mersenne Numbers and Generalized Fermat Numbers ) the primality for 10 million decimal numbers can be checked with a computer in some weeks or months.

To summarize: As long, as you can not prove mathematically that your algorithm produces a specific prime number of at least 10 million decimals, you have to check the candidates for primality. And that is not possible today for random - or as far as i can see it your - numbers in the 10 million decimals range with known algorithms.

Last fiddled with by TauCeti on 2003-12-08 at 09:27
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