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I have to test extremely huge numbers (100k+ digits) in a reasonable time if they are probable primes. Are there codes for doing that? Are there C-codes interfaced with SAGE? Or something else?

I have to test the primality of some very big numbers (100k+ digits), I mean it is enough to know if they are probable primes. I would know if there is a program in C for doing that. And what C-compiler do you suggest? Are there some C-libraries which allow probable primes tests?

f(n,p)={d=ceil(log(2)/log(10)(n-1));s=lift(Mod(2,p)^(n-1));t=lift(Mod(10,p)^d);‌​u=lift(Mod((2s-1)*t+s-1,p));u}

v=[100000..101000]

forprime(q=1,10^7,z=select(m->f(m,q)==0,v);if(length(z)>0,v=setminus(v,z);print(‌​q," ",length(z)," ",length(v))))

This is a program for PARI. For numbers of the form (2^k-1)*10^d+2^(k-1)-1 where d is the number of decimal digits of 2^(k-1)-1 in the range k=[100000..101000], it displays numbers with no factor below 10^7. Can somebody translate this PARI program in a SAGE program?

I have to find the smallest factor of a big number with SAGE. The problem with the factor command is that it displays the results only when the number is fully factored and so for a big number it could take an eternity to have the result. Has somebody a good program for finding factors of a big number without waiting for a full factorization?

@dan_fulea and what if I want to cancel the factorization?

@dan_fulea the program checks primes or probable primes? I am looking for probable primes.

@Emmanuel Charpentier no it is a project in mathexchange.

The object is to find all integers k , lets say in the range [-10^9,10^9], for which the number 40!+k splits in three prime factors with 16 decimal digits. Has somebody an efficient routine for Sage?

The ec(k) numbers are so defined: ec(k)=(2^k-1)10^d+2^(k-1)-1, where d is the number of decimal digits of 2^(k-1)-1. Examples of these numbers are: 31, 157, 3115, 40952047,... I found that up to k=565.000 there is no prime of the form (2^k-1)10^d+2^(k-1)-1 which is congruent to 6 mod 7, so I conjectured that there is no prime of this form congruent to 6 mod 7. Has somebody a program for Sage for checking this conjecture further?

The ec numbers are so defined:

ec(k) = (2^k-1)*10^d + 2^(k-1) - 1

where d is the number of decimal digits of 2^(k-1) - 1 . In other words these numbers are formed by the base 10 concatenation of two consecutive Mersenne numbers, for example: 157, 12763, 40952047...

For some values of k, ec(k) is probable prime. I found that up to k=565.000 there is no probable prime of the form (2^k-1)*10^d + 2^(k-1) - 1 which is congruent to 6 mod 7. So I conjectured that there is no probable prime of this form congruent to 6 mod 7. Has somebody an efficient program for Sage to test this conjecture further?