what is the best primality software?

[bbb120]

Originally Posted by VBCurtis

A "bug" is unexpected behavior. Using a primality test that isn't known to be correct on your input is not a bug.

You don't understand the software, and you're mistaken about having found a bug. Maybe ease up on the accusations until you understand what the various pfgw flags and tests do?

but the number being tested does not meet the conditions that are necessary for the test output to be valid

not meet the conditions

what conditions?
the pfgw report 999998912894617 as prime with -tp options,and it is not a bug??????????

[VBCurtis]

The post you keep quoting states the exact condition that you keep questioning. I've re-quoted it here so you can read the last line of the post. Is your number of the form k * b^n-1? That format assumes k < b^n, I think, but that may not be relevant.

Originally Posted by rogue

Before you do a primality test you should search for factors then run a PRP test.
I see that -tp shows prime, but the number being tested does not meet the conditions that are necessary for the test output to be valid.

-tm is used for numbers of the form k*b^n+1. -tp is used for numbers of the form k*b^n-1.

[bbb120]

Originally Posted by VBCurtis

The post you keep quoting states the exact condition that you keep questioning. I've re-quoted it here so you can read the last line of the post. Is your number of the form k * b^n-1? That format assumes k < b^n, I think, but that may not be relevant.

even 999998912894617 is not the form k * b^n-1, it should not tell me that it is prime.
it can tell me it is lucas prp at most!

[bbb120]

https://www.alpertron.com.ar/ECM.HTM
we can test primality use this website,it is very easy to use!

[VBCurtis]

Originally Posted by bbb120

even 999998912894617 is not the form k * b^n-1, it should not tell me that it is prime.
it can tell me it is lucas prp at most!

We, the humans on this forum, are telling you not to use that test flag -tp on that number. If you do, you're not using the software as directed so the results are not reliable. You've now confirmed, repeatedly, that you have an example where it is not reliable. So what? Don't use PFGW with -tp unless you have a number of the specified form.

Please stop whining that the software doesn't work. Garbage in, garbage out.

[bbb120]

Originally Posted by VBCurtis

We, the humans on this forum, are telling you not to use that test flag -tp on that number. If you do, you're not using the software as directed so the results are not reliable. You've now confirmed, repeatedly, that you have an example where it is not reliable. So what? Don't use PFGW with -tp unless you have a number of the specified form.

Please stop whining that the software doesn't work. Garbage in, garbage out.

Timing[PrimeQ[2^116224 - 15905]]
{427.864, True}
34987 decimal digits
mathematica use 427.864s to test primality on 2^116224 - 15905(34987 decimal digits)

PFGW only use 26.5976s!!!!!!!!

427.864/26.5976=16.0865642013

why pfgw is much faster than mathematica???????

PFGW Version 4.0.3.64BIT.20220704.Win_Dev [GWNUM 29.8]

Resuming input file input2 at line 2

Primality testing 2^116224-15905 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 31, base 1+sqrt(31)
2^116224-15905 is Fermat and Lucas PRP! (26.5976s+0.0008s)

Done.

[kriesel]

Originally Posted by bbb120

why pfgw is much faster than mathematica???????

PFGW Version 4.0.3.64BIT.20220704.Win_Dev [GWNUM 29.8]

https://mathematica.stackexchange.co...in-mathematica
(Found trivially easily and quickly, first link returned by DuckDuckGo for search string "does mathematica use gwnum".)

[Dr Sardonicus]

Originally Posted by bbb120

<snip>

pfgw -b2 input3.txt
***WARNING! file input3.txt may have already been fully processed.

80383745745363949125......7534901 is 2-PRP! (0.0021s+0.0001s)

<snip>

pfgw -b14 input3.txt

***WARNING! file input3.txt may have already been fully processed.

80383745745363......72597534901 is 14-PRP! (0.0016s+0.0001s)

but
14^((n-1)/4)mod n not equal ±1
and 14^((n-1)/2)mod n equal 1 (not equal -1),
thus,miller rabin tell us that n must be a composite number
but,pfgw tell us n is 14-PRP
so pfgw must use fermat test as "prp test",not miller-rabin.

Please explain the warning about input3.txt

FWIW I verified that n "passes" the Fermat PRP tests to base 2 and base 14.

When n tests as a Fermat PRP but as composite by Miller-Rabin to the same base, a factorization is immediately obtainable.

In the case at hand, we have the factors

p=2004791083197498027091532260422734265025940830205662543872531023690016085350598121358111595798609866791081582542679083484572616906958584643763990222898400226296015918301

and q = 2*p - 1,

q = 4009582166394996054183064520845468530051881660411325087745062047380032170701196242716223191597219733582163165085358166969145233813917169287527980445796800452592031836601

EDIT:

Originally Posted by bbb120

I cannot understand this sentence very well!

It means you've been using the wrong tool for the job. And then you've been complaining that the problem is that the tool is broken.

[kar_bon]

Originally Posted by Dr Sardonicus

***WARNING! file input3.txt may have already been fully processed.

Please explain the warning about input3.txt

It's only a warning from pfgw: if pfgw runs any file an *.ini file is created where you can find the last processed input file. Delete the .ini and there is no warning. So nothing to worry sbout.

[mathwiz]

Originally Posted by VBCurtis

We, the humans on this forum, are telling you not to use that test flag -tp on that number. If you do, you're not using the software as directed so the results are not reliable. You've now confirmed, repeatedly, that you have an example where it is not reliable.

“On two occasions, I have been asked [by members of Parliament], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able to rightly apprehend the kind of confusion of ideas that could provoke such a question.”

[bbb120]

Originally Posted by paulunderwood

./pfgw64 -tc -q2887148238050771212671429597130393991977609459279722700926516024197432303799152733116328983144639225941977803110929349655578418949441740933805615113979999421542416933972905423711002751042080134966731755152859226962916775325475044445856101949404200039904432116776619949629539250452698719329070373564032273701278453899126120309244841494728976885406024976768122077071687938121709811322297802059565867
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]

Primality testing 2887148238050771212671429597130393991977609459279722700926516024197432303799152733116328983144639225941977803110929349655578418949441740933805615113979999421542416933972905423711002751042080134966731755152859226962916775325475044445856101949404200039904432116776619949629539250452698719329070373564032273701278453899126120309244841494728976885406024976768122077071687938121709811322297802059565867 [N-1/N+1, Brillhart-Lehmer-Selfridge]                                    
Running N-1 test using base 2                                                  
Factored: 10475096971045985224204423648945582453962513105348124302901261662540724079869634880456766224539126779375883658239075983560088580357347                                    
2887148238050771212671429597130393991977609459279722700926516024197432303799152733116328983144639225941977803110929349655578418949441740933805615113979999421542416933972905423711002751042080134966731755152859226962916775325475044445856101949404200039904432116776619949629539250452698719329070373564032273701278453899126120309244841494728976885406024976768122077071687938121709811322297802059565867 is composite (0.0115s+0.0011s)

There are no known composite numbers that pass PFGW's Fermat+Lucas which can be achieved by running the combined N+1/N-1 tests with the switch -tc.

some composite number(for example Carmichael number) may pass Fermat+Lucas test,
and fermat test number can not identify it(except base is its factor),but miller rain can,
so miller rabin can improve the reliable of the software ,
Fermat+Lucas test can not improve any reliable at present!