how can I find nextprime with the help of pfgw?

[bbb120]

I want to know nextprime(10^1000),

pfgw -qnextprime(10^1000)

but pfgw tell me

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

nextprime(10^1000) is trivially prime!: 2

Done.

but what I really want to get is 10^1000+453 ,not nextprime(10^1000)
who can help me?

[a1call]

Read the ABCFILEFORMAT file (or something similar)
And script a variable to increment with logical steps say by 2. There are no (simple) shortcuts. You will have to sieve (software does a basic sieve) and check each increment for primality.
You should also look into Pari-GP. It’s a programmable calculator and will give you next/previous primes.
I combine both software for my hobby. PFGW is the fastest primality checking software, but is not as flexible as PARI-GP.
Just my 2 cents.

[bbb120]

What is the smallest prime number greater than F15？
F15=2^(2^15)+1 fermat number
I check F15+n for 0<=n<=80000,
but I can not find any prp,
but prime number theorem tells us that the gap is approximate 2^15*log(2)=22713.0468126，
80000/22713=3.5222119491

my input.txt content

ABC2 $a+(2^(2^15)+1)
a: from 00000 to 80000 step 2

my command is

pfgw input.txt -b2

I use pfgw to search the prp

[LaurV]

2^15 is 32768, so those numbers have about 10 thousand digits, so it is not difficult to PRP them, even pari/gp can do it quite fast. How about you first learn how to use pfgw, before opening a gazilion threads claiming "bugs" in all available tools we use?

[bbb120]

Originally Posted by LaurV

2^15 is 32768, so those numbers have about 10 thousand digits, so it is not difficult to PRP them, even pari/gp can do it quite fast. How about you first learn how to use pfgw, before opening a gazilion threads claiming "bugs" in all available tools we use?

I find the prp 118112+(2^(2^15)+1) !
F15+118112

check command

pfgw -q118112+(2^(2^15)+1) -tc

output

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

Primality testing 118112+(2^(2^15)+1) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 19, base 9+sqrt(19)
118112+(2^(2^15)+1) is Fermat and Lucas PRP! (4.9587s+0.0006s)

Done.

check command

pfgw -q118112+(2^(2^15)+1) -tp

output

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

Primality testing 118112+(2^(2^15)+1) [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 7+sqrt(7)
118112+(2^(2^15)+1) is Lucas PRP! (4.0007s+0.0006s)

Done.

why N+1 test base is different?
one is base 9+sqrt(19), the other is base 7+sqrt(7)

[bbb120]

Originally Posted by LaurV

2^15 is 32768, so those numbers have about 10 thousand digits, so it is not difficult to PRP them, even pari/gp can do it quite fast. How about you first learn how to use pfgw, before opening a gazilion threads claiming "bugs" in all available tools we use?

pfgw is really a good primality software !

[bbb120]

Originally Posted by bbb120

I find the prp 118112+(2^(2^15)+1) !
F15+118112

check command

pfgw -q118112+(2^(2^15)+1) -tc

output

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

Primality testing 118112+(2^(2^15)+1) [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 7
Running N+1 test using discriminant 19, base 9+sqrt(19)
118112+(2^(2^15)+1) is Fermat and Lucas PRP! (4.9587s+0.0006s)

Done.

check command

pfgw -q118112+(2^(2^15)+1) -tp

output

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

Primality testing 118112+(2^(2^15)+1) [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 7, base 7+sqrt(7)
118112+(2^(2^15)+1) is Lucas PRP! (4.0007s+0.0006s)

Done.

why N+1 test base is different?
one is base 9+sqrt(19), the other is base 7+sqrt(7)

I test primality on 118112+(2^(2^15)+1) with miller-rabin test with random bases,
and it tell me true true true!!!
my mathematica miller rabin code

(*miller rabin test,n0 big odd integer,a0 base*)
MillerRabin[n0_,a0_]:=Module[{n=n0,a=a0,s,m,t1,k},
    s=0;m=n-1;While[Mod[m,2]==0,m=m/2;s=s+1];
    t1=PowerMod[a,m,n];
    If[t1==1,Return[True]];
    k=0;While[k<s-1&&t1!=n-1,k=k+1;t1=Mod[t1^2,n]];
    If[t1==n-1,Return[True],Return[False]]
]