I have a problem that I would like Mathematica could solve for me, but haven’t been able to do it. I thought about using the Table command, but that command would show millions of solutions (in case the computer has enough memory available) while I only want a very, very, small fraction of them.

The question is the following: I have 2 expressions (which may be called F(i,j,k) and G(i,j,k)) and I would like that Mathematica would test if F(i,j,k)=0 and G(i,j,k)=1 for all i, j and k belonging to the set {0,1,2…,728}. Then I would like to throw away those (i, j, k) such that both i, j and k are multiples of 28. Only the remaining interest me.

This is a question related to finite fields. I wish simply to know if a system of equations has solutions in the field of order 729=3^6 that are not in the smaller field of order 27=3^3.

I don’t know if Mathematica can do this. In spite of having to do 729 * 729 * 729 calculations (this would take days to compute?) the expected results should be manageable (several hundreds, at most). In case you are interested the expressions F(i,j,k) and G(i,j,k) are:

F(i,j,k)=
PolynomialMod[PolynomialMod[2 (w^i)^43 (w^j) + 2 (w^i)^30 (w^j)^14 + (w^i)^17 (w^j)^27 + (w^i)^4 (w^j)^40 + (w^ i)^39 (w^j)^4 (w^k) + (w^i)^13 (w^j)^30 (w^k) +  2 (w^j)^43 (w^k) + (w^i)^9 (w^j)^33 (w^k)^2 + (w^i)^31 (w^j)^10 (w^ k)^3 +2 (w^i)^5 (w^j)^36 (w^k)^3 + (w^i)^40 (w^k)^4 + (w^ i)^27 (w^j)^13 (w^k)^4 +(w^i) (w^j)^39 (w^k)^4 + 2 (w^i)^36 (w^j)^3 (w^k)^5 + 2 (w^i)^28 (w^j)^9 (w^k)^7 + (w^i)^33 (w^j)^2 (w^k)^9 +2(w^i)^7 (w^j)^28 (w^k)^9 + (w^i)^3 (w^j)^31 (w^k)^10 + (w^i)^30 (w^ j) (w^k)^13 + (w^i)^4 (w^j)^27 (w^k)^13 + 2 (w^j)^30 (w^k)^14 + (w^i)^27 (w^k)^17 + (w^i)^13 (w^j)^4 (w^ k)^27 + (w^j)^17 (w^k)^27 + 2 (w^i)^9 (w^j)^7 (w^k)^28 + 2 (w^i)^14 (w^k)^30 + (w^i) (w^j)^13 (w^k)^30 +(w^i)^10 (w^j)^3 (w^ k)^31 + (w^i)^2 (w^j)^9 (w^k)^33 + 2 (w^i)^3 (w^j)^5 (w^k)^36 + (w^i)^4 (w^j) (w^k)^39 + (w^j)^4 (w^ k)^40 + 2 (w^i) (w^k)^43, w^52-1],3]

G(i,j,k)=
PolynomialMod[PolynomialMod[ (w^i)^52 + (w^i)^39 (w^j)^13 + (w^i)^13 (w^j)^39 + (w^j)^52 +  2 (w^i)^48 (w^j)^3 (w^k) +  2 (w^i)^9 (w^j)^42 (w^k) + 
(w^i)^40 (w^j)^9 (w^k)^3 + 2 (w^i)^27 (w^j)^22 (w^k)^3 + 2 (w^i) (w^j)^48 (w^k)^3 + (w^i)^36 (w^j)^12 (w^k)^4 +(w^i)^28 (w^ j)^18 (w^k)^6 + 2 (w^i)^42 (w^j) (w^k)^9 +2 (w^i)^16 (w^j)^27 (w^k)^9 + (w^i)^3(w^j)^40 (w^k)^9 + (w^i)^12 (w^ j)^30 (w^k)^10 + (w^i)^30 (w^j)^10 (w^k)^12 + (w^i)^4 (w^j)^36 (w^ k)^12 + (w^i)^39 (w^k)^13 + (w^j)^39 (w^k)^13 + 2 (w^i)^27 (w^j)^9 (w^k)^16 + (w^i)^6 (w^j)^28 (w^k)^18 +  2 (w^i)^3 (w^j)^27 (w^k)^22 + 2 (w^i)^22 (w^j)^3 (w^k)^27 +  2 (w^i)^9 (w^j)^16 (w^k)^27 + (w^i)^18 (w^j)^6 (w^k)^28 + (w^ i)^10 (w^j)^12 (w^k)^30 + (w^i)^12 (w^j)^4 (w^k)^36 + (w^i)^13 (w^ k)^39 + (w^j)^13 (w^k)^39 + (w^i)^9 (w^j)^3 (w^k)^40 + 2 (w^i) (w^j)^9 (w^k)^42 +2 (w^i)^3 (w^j) (w^k)^48 + (w^k)^52, w^52-1],3]

One last question: I suppose that Mathematica orders (i, j, k) with lexicographical order. Do you know any command that asks Mathematica to compute the “first” (in the order it uses) 10.000 or 100.000 (or any other quantity) (i, j, k) and hours or days later continue from where it stopped?