final_mws_primes.m

load "fake_allcurves.m";

//we store the necessary MWS primes necessary_pr with respect to each element omega in Omega for each curve
//we consolidate this information in necessary_MWS_primes, which stores the necessary MWS primes for each of the curves in the set. 
//This information can the read from json.

//we have data from Sage concerning three primes of good ordinary reduction, done_pr records which subset of these primes 
//is necessary for a successful sieve together with the auxiliary integer. 

//we consolidate the auxiliary integers data from done_pr 
//we consolidate the data from done_pr in ordinary_primes_modulus
//the modulus can be extracted from these data and is recorded in json

necessary_pr := [
    [
        [ 3, 61 ],
        [ 3, 61 ],
        [ 61 ],
        [ 3, 61 ],
        [ 3, 61 ],
        []
    ],
    [
        [ 23 ],
        [ 23 ],
        [ 23 ],
        [ 23 ]
    ],
    [
        [ 13 ],
        [ 13 ],
        [ 13, 107, 179, 673 ],
        [ 13, 107 ],
        [ 13 ],
        [ 13, 107 ],
        [ 13 ]
    ],
    [
        [ 1873 ],
        [ 1873 ],
        [ 1873, 2857, 2897 ]
    ],
    [
        [],
        [ 11, 19, 31 ],
        [ 11, 19 ],
        [],
        [ 11, 19 ],
        [ 3, 11, 19, 31 ]
    ],
    [
        [],
        [ 5 ],
        [ 5 ],
        [],
        [],
        [],
        []
    ],
    [
        [ 223, 443 ]
    ],
    [
        [ 359, 443, 2129 ],
        [ 359, 443, 2129, 19, 43, 263 ],
        [ 359, 443, 19, 43, 331 ],
        [ 359, 443, 2129 ],
        [ 359, 443, 2129 ],
        [ 359, 443, 2129, 19, 43 ],
        [ 359, 443, 2129 ],
        [ 359, 443, 2129, 19, 43, 89 ],
        [ 359, 443, 2129, 19, 43, 89 ]
    ],
    [
        [],
        [],
        [ 29, 59 ],
        [],
        [ 29, 59 ],
        [],
        [],
        [],
        [ 29, 59 ],
        [],
        [],
        [ 3, 29, 59 ],
        [],
        [],
        [ 29, 59 ],
        [],
        [],
        [ 3, 29, 59, 107 ],
        [],
        [],
        [ 29, 59 ]
    ],
    [
        [ 251, 487 ],
        [ 251, 487 ],
        [ 251, 487 ],
        [ 251 ],
        [],
        [ 251, 487 ],
        [],
        [],
        [],
        [ 251 ],
        [ 251 ],
        [ 251 ],
        [],
        [],
        [],
        [ 251, 487 ],
        [ 251 ],
        [],
        [ 251, 487 ],
        [ 487 ],
        [ 251 ],
        [ 251 ],
        [ 487, 1801 ],
        [ 251, 487 ],
        [ 251 ],
        [ 251 ],
        [ 251, 487 ]
    ],
    [
        [],
        [ 19, 647 ],
        [ 19 ],
        [ 19 ]
    ],
    [
        [ 251 ],
        [ 251, 541 ],
        [],
        [ 251 ],
        [ 251 ],
        [ 251 ],
        [ 251 ],
        []
    ],
    [
        [],
        [ 71 ],
        [ 71, 3037 ],
        [ 71 ],
        [ 71, 2399 ],
        [ 71 ],
        [ 71 ],
        [ 71, 2399, 3037 ],
        [ 71 ],
        [ 71 ],
        [ 71, 2399, 3037 ],
        [],
        [ 71, 3037 ],
        [ 71, 2399 ],
        [ 71, 2399 ],
        [ 71, 3037 ],
        [ 71 ],
        [ 71, 3037 ],
        [ 71 ],
        [ 71 ],
        [ 71, 2399, 3037, 5531 ],
        [ 71 ],
        [ 71, 2399, 3037 ],
        [],
        [ 71 ],
        [],
        [ 71, 3037 ]
    ],
    [
        [ 71, 1873 ],
        [ 71, 1873 ],
        [ 71, 1873 ],
        [ 71, 1873 ],
        [],
        [ 71, 1873 ]
    ],
    [
        [],
        [ 3 ],
        [ 3 ],
        [],
        [ 3 ],
        [ 3 ],
        [],
        [],
        []
    ],
    [
        [ 4733, 6689 ],
        [ 17, 167, 4733, 6689 ],
        [ 17, 167 ],
        [ 17, 167, 6689 ],
        [ 17 ],
        [ 17 ]
    ],
    [
        [],
        [],
        [],
        [ 17 ],
        [ 17 ],
        [ 17, 103, 457, 547 ],
        [ 17 ],
        [ 17, 103 ],
        [ 17 ],
        [ 17 ],
        [ 17 ],
        [],
        [],
        [],
        [],
        [],
        [],
        [],
        [ 17, 103 ],
        [ 17 ],
        [ 17, 103 ]
    ],
    [
        [],
        [ 3, 5, 137, 167 ],
        [ 3 ],
        [ 3, 5, 137 ],
        [],
        [],
        []
    ],
    [
        [ 3 ],
        [ 3 ],
        [],
        [ 3, 11 ],
        [ 3, 11, 83, 139 ],
        [ 3 ],
        [],
        [ 3, 11, 61 ],
        [ 3, 11 ],
        [],
        [ 3 ],
        [ 3, 11 ],
        [ 3, 11 ],
        [ 3, 11, 139 ],
        [],
        [ 3, 11 ]
    ],
    [
        [ 3, 7, 181, 233 ],
        [ 3, 7 ],
        [ 3 ],
        [ 3 ],
        [ 3, 7 ],
        [ 3, 7, 181 ],
        [ 3 ],
        [ 3, 7 ],
        [ 3, 7 ],
        [ 3, 7 ],
        [ 3 ],
        [],
        [ 3, 7 ],
        [ 3, 7, 181, 233 ],
        [ 3 ],
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        [ 3 ],
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        [ 3, 7, 181 ],
        [],
        [ 3, 7 ],
        [ 3 ],
        [],
        [ 3 ],
        [ 3, 7 ],
        [ 3, 7, 181 ],
        [ 3 ]
    ],
    [
        [ 41, 43 ],
        [ 41, 43 ],
        [ 41, 43 ],
        [ 41 ],
        [ 41 ],
        [ 41, 43, 251 ],
        [],
        [ 41, 43 ],
        [ 41, 43, 251 ],
        [],
        [],
        [],
        [ 41, 43, 101 ],
        [ 41, 43, 251 ],
        [ 41 ],
        [],
        [],
        []
    ],
    [
        [],
        [ 3 ],
        [ 3, 541 ],
        [ 3, 83, 823 ],
        [ 3, 83 ],
        [],
        [ 3, 823 ],
        [],
        [ 83 ]
    ],
    [
        [ 167, 1613, 5711, 5981 ],
        [ 43, 167, 1613, 5711, 5981 ],
        [ 43, 167, 1613, 5711 ],
        [ 167, 1613, 5711, 5981 ],
        [],
        [ 43, 167, 1613, 5711 ],
        [ 43, 167, 1613, 5711, 5981 ],
        [],
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        [ 167, 1613, 5711, 5981 ],
        [ 43, 167, 1613 ],
        [ 43 ],
        [],
        [ 43, 167, 1613 ],
        [ 43, 167, 1613, 5711 ],
        [],
        [ 43, 167, 1613, 5711, 5981 ],
        [ 167, 1613, 5711, 5981 ]
    ],
    [
        [],
        [ 3 ],
        [],
        [ 3 ],
        [],
        [],
        []
    ],
    [
        [],
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        [],
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        [],
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        [],
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        [],
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        [],
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        [ 7 ],
        [],
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        [],
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        [],
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        [ 97 ],
        [],
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    ],
    [
        [ 23 ],
        [],
        [ 19 ],
        [],
        [ 23 ],
        [ 19, 23 ]
    ],
    [
        [],
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        [],
        [],
        [ 67 ],
        []
    ],
    [
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        [],
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    [
        [ 3, 7 ],
        [],
        [],
        [],
        [ 3, 97, 461, 983 ]
    ],
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        [ 503, 127, 503 ],
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        [],
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        [],
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        [],
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        [ 503, 127, 503, 1033, 1321 ],
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        [],
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    ],
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        [],
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        [],
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        [],
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    ],
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        [],
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// for each omega in Omega, we record the primes of good ordinary reduction used 
//in the modulus, while the last entry is the auxiliary integer
//if done_pr[i][j] it means that for the j-th element of the i-th curve no sieving
//was necessary

done_pr := 
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//we consolidate the auxiliary integers data from done_pr 
// auxiliary_integers:= [0 : i in [1..#done_pr]];

// for i in [1..#done_pr] do
//     i;
//     auxiliary_integers[i] := LCM([S[#S] : S in done_pr[i] | S[#S] gt 0] cat [1]);
// end for;

// #[i : i in [1..#done_pr] | auxiliary_integers[i] eq 2];
// #[i : i in [1..#done_pr] | auxiliary_integers[i] eq 3];
// #[i : i in [1..#done_pr] | auxiliary_integers[i] eq 4];


auxiliary_integers := [ 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 
1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2,
1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1,
1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1,
1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1,
2, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2,
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1,
1, 1, 1, 1 ];

//we consolidate the data from done_pr 
// ordinary_primes_modulus := [[]: i in [1..#done_pr]];

// for i in [1..#done_pr] do
//     i;
//     for S in done_pr[i] do
//         ordinary_primes_modulus[i] := ordinary_primes_modulus[i] cat Prune(S);
//         ordinary_primes_modulus[i] := SetToSequence(SequenceToSet(ordinary_primes_modulus[i]));
//     end for;
// end for;



ordinary_primes_modulus := [
    [ 5, 7 ],
    [ 11, 5 ],
    [ 13, 17 ],
    [ 11, 7 ],
    [ 11, 3 ],
    [ 3, 5 ],
    [ 5, 17 ],
    [ 5, 17 ],
    [ 3, 7 ],
    [ 5, 7 ],
    [ 11, 13 ],
    [ 5, 7 ],
    [ 23, 5 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 11, 5 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 13, 5 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 11, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 11, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 3, 7 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 11, 3 ],
    [ 5, 17 ],
    [ 5, 7 ],
    [ 11, 5, 7 ],
    [ 23, 17 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 13, 19 ],
    [ 13, 7 ],
    [ 11, 3 ],
    [ 11, 7 ],
    [ 11, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 3, 5 ],
    [ 11, 5 ],
    [ 3, 7 ],
    [ 11, 7 ],
    [ 11, 17 ],
    [ 17, 7 ],
    [ 11, 5 ],
    [ 11, 7 ],
    [ 11, 13 ],
    [ 13, 3 ],
    [ 13, 17 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 11, 3 ],
    [ 5, 7 ],
    [ 11, 19 ],
    [ 13, 7 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 11, 5 ],
    [ 11, 5 ],
    [ 3, 7 ],
    [ 3, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 11, 17 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 13, 7 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 13, 5 ],
    [ 11, 13 ],
    [ 11, 3 ],
    [ 11, 17 ],
    [ 17, 7 ],
    [ 13, 17 ],
    [ 11, 13 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 11, 5 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 3, 7 ],
    [ 13, 7 ],
    [ 11, 5 ],
    [ 13, 7 ],
    [ 3, 5 ],
    [ 13, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 17, 7 ],
    [ 13, 3 ],
    [ 11, 13 ],
    [ 5, 7 ],
    [ 17, 19 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [],
    [ 3, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 3, 7 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 11, 13 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 13, 19 ],
    [ 13, 7 ],
    [ 13, 7 ],
    [ 11, 7 ],
    [ 11, 7 ],
    [ 11, 3 ],
    [ 11, 13 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 7, 19 ],
    [ 11, 7 ],
    [ 13, 3 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 11, 7 ],
    [ 11, 5, 7 ],
    [ 3, 5 ],
    [ 11, 19 ],
    [ 13, 17 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 11, 5 ],
    [ 11, 7 ],
    [ 13, 5, 17 ],
    [ 3, 5 ],
    [ 3, 7 ],
    [ 23, 19 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 11, 13 ],
    [ 3, 5 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 11, 3 ],
    [ 23, 17 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 11, 7 ],
    [ 13, 17, 19 ],
    [ 13, 17 ],
    [ 13, 17 ],
    [ 13, 7 ],
    [ 17, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 11, 3 ],
    [ 3, 5 ],
    [ 11, 13, 19 ],
    [ 5, 7 ],
    [ 11, 13 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 3, 5 ],
    [ 3, 7 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 3, 7 ],
    [ 5, 7 ],
    [ 3, 7 ],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 11, 13 ],
    [ 13, 7 ],
    [ 3, 5 ],
    [ 5, 17 ],
    [],
    [ 11, 7 ],
    [ 3, 7 ],
    [ 5, 17 ],
    [ 11, 5 ],
    [ 13, 17 ],
    [ 11, 13 ],
    [ 17, 41 ],
    [ 5, 7 ],
    [ 11, 5 ],
    [ 11, 31 ],
    [ 13, 7 ],
    [ 3, 5 ],
    [ 13, 19 ],
    [ 11, 13 ],
    [ 3, 7 ],
    [ 5, 7 ],
    [],
    [ 5, 7 ],
    [ 3, 7 ],
    [ 3, 5 ],
    [ 13, 17 ],
    [ 13, 5 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 11, 3 ],
    [ 13, 7 ],
    [ 13, 17 ],
    [ 5, 17 ],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 23, 17 ],
    [ 5, 7 ],
    [ 13, 17 ],
    [ 13, 5, 17 ],
    [ 3, 5 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 11, 13, 19 ],
    [ 17, 7 ],
    [ 5, 19 ],
    [ 11, 7 ],
    [ 3, 5 ],
    [ 11, 13 ],
    [ 5, 7 ],
    [ 13, 5 ],
    [ 3, 5 ],
    [ 13, 19 ],
    [ 23, 17 ],
    [],
    [ 5, 7 ],
    [ 13, 7 ],
    [],
    [ 3, 7 ],
    [ 11, 17 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [ 13, 7 ],
    [],
    [ 11, 13 ],
    [ 13, 17 ],
    [ 5, 7 ],
    [ 13, 3 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 11, 13 ],
    [ 11, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 13, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 3, 19 ],
    [ 5, 7 ],
    [ 11, 3 ],
    [ 5, 7 ],
    [ 11, 3 ],
    [ 11, 5 ],
    [ 3, 5 ],
    [ 5, 17 ],
    [ 11, 7 ],
    [ 13, 17 ],
    [ 13, 7 ],
    [ 3, 5 ],
    [ 5, 7 ],
    [],
    [ 5, 7 ],
    [ 11, 7 ],
    [ 11, 7 ],
    [ 11, 5 ],
    [ 5, 7 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 3, 7 ],
    [ 13, 7 ],
    [ 5, 7 ],
    [ 3, 7 ],
    [ 13, 19 ],
    [ 13, 17 ],
    [ 5, 7 ],
    [ 3, 5 ],
    [ 3, 5 ],
    [ 3, 7 ],
    [ 3, 5 ],
    [],
    [ 13, 17, 7 ],
    [ 13, 7 ]
];

//#[i : i in [1..#ordinary_primes_modulus] |ordinary_primes_modulus[i] eq [] ];  //8 = 6+2 (pointless)


//we consolidate the data from necessary_pr
// necessary_MWS_primes := [[]: i in [1..#necessary_pr]];

// for i in [1..#necessary_pr] do
//     i;
//     for S in necessary_pr[i] do
//         necessary_MWS_primes [i] := necessary_MWS_primes [i] cat S;
//         necessary_MWS_primes [i] := Sort(SetToSequence(SequenceToSet(necessary_MWS_primes[i])));
//     end for;
// end for;

necessary_MWS_primes := [
    [ 3, 61 ],
    [ 23 ],
    [ 13, 107, 179, 673 ],
    [ 1873, 2857, 2897 ],
    [ 3, 11, 19, 31 ],
    [ 5 ],
    [ 223, 443 ],
    [ 19, 43, 89, 263, 331, 359, 443, 2129 ],
    [ 3, 29, 59, 107 ],
    [ 251, 487, 1801 ],
    [ 19, 647 ],
    [ 251, 541 ],
    [ 71, 2399, 3037, 5531 ],
    [ 71, 1873 ],
    [ 3 ],
    [ 17, 167, 4733, 6689 ],
    [ 17, 103, 457, 547 ],
    [ 3, 5, 137, 167 ],
    [ 3, 11, 61, 83, 139 ],
    [ 3, 7, 181, 233 ],
    [ 41, 43, 101, 251 ],
    [ 3, 83, 541, 823 ],
    [ 43, 167, 1613, 5711, 5981 ],
    [ 3 ],
    [ 7, 13, 71, 79, 89, 3413 ],
    [ 97, 281, 347, 521, 829 ],
    [ 19, 23 ],
    [ 23, 67, 241 ],
    [ 11, 19, 67, 71 ],
    [ 3, 7, 97, 461, 983 ],
    [ 127, 503, 1033, 1291, 1321 ],
    [ 61, 197 ],
    [ 19, 23, 191, 283, 331, 661, 2719, 2797, 3203 ],
    [ 3, 173, 179 ],
    [ 107, 139 ],
    [ 3, 19, 43, 79, 107 ],
    [ 19, 47, 239, 823, 1033 ],
    [ 23, 137, 157 ],
    [ 3, 5, 29, 59, 97, 251 ],
    [ 521, 5477 ],
    [ 19, 61, 71, 167 ],
    [ 17, 151, 167, 1171, 3257, 4259 ],
    [ 19, 683, 5953 ],
    [ 67, 79, 107, 1823, 2693, 2729, 4789 ],
    [ 83, 241, 883, 1601, 1619 ],
    [ 3, 19, 257 ],
    [ 11, 331, 809, 821, 907, 1217 ],
    [ 7, 67, 271, 743, 2131 ],
    [ 13, 17, 103, 401 ],
    [ 137 ],
    [ 223, 599, 1901, 6449, 9157 ],
    [ 3 ],
    [ 3, 7, 41 ],
    [ 79, 103, 1543, 8009 ],
    [ 67, 5569 ],
    [ 3, 7, 197 ],
    [ 7, 541, 2917, 4933 ],
    [ 23, 71, 1733, 1873, 6287 ],
    [ 331, 661, 1319 ],
    [ 359, 1087, 2341, 2741, 3889 ],
    [ 3, 19, 103, 239, 503, 601, 733 ],
    [ 19, 71, 193, 197, 227, 487, 1193, 8363 ],
    [ 23, 31, 139 ],
    [ 37, 491, 587, 3469 ],
    [ 5 ],
    [ 3, 19 ],
    [ 103, 163, 167, 3559, 4933, 6361 ],
    [ 181, 197, 239 ],
    [ 19, 41, 197, 1783 ],
    [ 5, 47, 59 ],
    [ 79, 4177, 9323 ],
    [ 3, 29, 31, 107 ],
    [ 11, 19, 2687, 4051, 7691, 9679 ],
    [ 71, 89, 1277, 1429 ],
    [ 29, 31, 103, 193, 227, 4423, 8093 ],
    [ 127, 523, 3209 ],
    [ 17, 563, 3907 ],
    [ 19, 4007, 9619 ],
    [ 19, 41, 257 ],
    [ 9007, 9859 ],
    [ 5, 109 ],
    [ 487 ],
    [ 17, 23, 6689 ],
    [ 5, 13, 59 ],
    [ 19, 1759, 2063, 3011 ],
    [ 239, 379, 457, 607, 3313, 5281 ],
    [ 109, 409, 1193, 1997 ],
    [ 3, 5, 1031 ],
    [ 47, 61, 97, 163, 349, 521, 683, 1699 ],
    [ 107, 1033, 1061, 1291, 1601, 3359, 4339, 4813 ],
    [ 43, 227, 263, 659, 7589, 7699 ],
    [ 11, 41 ],
    [ 67, 5569, 7741 ],
    [ 13, 41, 269, 457, 719 ],
    [ 13, 41, 107, 269 ],
    [ 281, 331, 557 ],
    [ 19, 149, 677, 827, 1031 ],
    [ 11, 149, 587 ],
    [ 3, 5, 7, 37, 107, 149 ],
    [ 7, 149, 197 ],
    [ 13, 41, 373, 1033, 1051, 1237, 1861, 3041 ],
    [ 31, 43, 103, 149, 1301, 2539, 7927 ],
    [ 7, 241 ],
    [ 107, 109, 191, 257 ],
    [ 11, 101, 347, 1663, 2383 ],
    [ 167 ],
    [ 29, 37, 157, 619, 2659 ],
    [ 271, 827, 2939 ],
    [ 127, 191, 443, 461, 2129, 3203 ],
    [ 43, 467, 1933, 3187 ],
    [ 47, 557, 947 ],
    [ 3, 59 ],
    [ 71, 1429 ],
    [ 659, 1453, 4091, 7001 ],
    [ 677 ],
    [ 19, 683, 853 ],
    [ 157, 1433, 1973, 3301 ],
    [ 73, 127, 151, 599, 769 ],
    [ 43, 97 ],
    [ 83, 137, 397, 1987 ],
    [ 3, 5, 7, 37, 41 ],
    [ 293, 673 ],
    [ 73, 137, 233, 541 ],
    [ 13, 17, 103, 269 ],
    [ 3 ],
    [ 11, 271, 277, 347, 1669 ],
    [ 37, 313 ],
    [ 47, 97, 3001, 3361 ],
    [ 31, 2423, 2711, 4871 ],
    [ 17, 277 ],
    [ 3, 19 ],
    [ 29, 449, 673, 1153, 2113, 2797, 5807 ],
    [ 127, 151, 199, 739, 1187, 1301 ],
    [ 83, 163, 229 ],
    [ 11, 113, 157, 239 ],
    [ 31, 109, 349, 499, 1801, 3343, 6719 ],
    [ 17, 19, 89 ],
    [ 239, 317, 487, 577, 6143 ],
    [ 7, 83, 103, 149, 197 ],
    [ 43, 701, 2179, 2311, 4003 ],
    [ 5, 29, 79, 197, 239 ],
    [ 41, 167, 1291, 2557, 2851, 3853 ],
    [],
    [ 7, 11, 19, 89 ],
    [ 17, 157, 197, 433, 1033 ],
    [ 47, 101 ],
    [ 151 ],
    [ 241 ],
    [ 149, 677, 5443 ],
    [ 23, 277, 1733, 1873 ],
    [ 5, 7 ],
    [ 227, 257, 271, 5381 ],
    [ 3, 19 ],
    [ 3, 17, 263 ],
    [ 1831, 2029, 4217, 8233 ],
    [ 79, 149, 4157, 4871 ],
    [ 3, 7 ],
    [ 31, 107, 139, 311, 313, 739, 997, 1423, 1973, 3517, 4339 ],
    [ 41, 457, 733, 829, 1291, 2687 ],
    [ 1669, 2311, 5261 ],
    [ 7, 89 ],
    [ 233, 653, 1217, 1669, 4441 ],
    [ 113, 127 ],
    [ 109, 127, 971, 2539, 5717, 8069 ],
    [ 37, 1019, 1103 ],
    [ 5 ],
    [ 3, 137 ],
    [ 431, 641, 3691, 4159, 4289, 8233 ],
    [ 347, 349, 373 ],
    [ 17, 41 ],
    [ 19, 89, 137 ],
    [ 131, 257 ],
    [ 19, 197, 659 ],
    [ 241, 487, 647, 811 ],
    [ 3, 19 ],
    [ 163, 223, 349, 431, 449, 1429 ],
    [ 41 ],
    [ 191, 283, 2473 ],
    [ 467, 997, 1009, 1481, 1783, 4363, 7243 ],
    [ 19 ],
    [ 11 ],
    [ 41, 661, 751 ],
    [ 43, 47, 131, 211 ],
    [ 3, 11, 73 ],
    [ 3, 53, 103 ],
    [ 3, 13, 41, 127, 241, 311 ],
    [ 3, 5, 17 ],
    [ 83, 97, 101, 1021, 2927 ],
    [ 47, 233, 331, 617, 1217 ],
    [ 139, 191, 5711 ],
    [ 3 ],
    [ 17, 823 ],
    [ 31, 89, 331, 839, 967, 1699, 2179, 2971, 8819 ],
    [ 101, 179, 277 ],
    [ 5 ],
    [ 4111, 4327, 7013 ],
    [ 3, 7, 17, 31 ],
    [ 23, 613, 967 ],
    [ 17, 83, 103 ],
    [ 47, 109, 389, 503 ],
    [ 367, 607, 659, 1973, 4159, 4561 ],
    [ 181, 1291 ],
    [ 3, 107, 233 ],
    [ 199, 463 ],
    [ 13, 421, 569, 2237, 5821 ],
    [ 41, 593, 2053 ],
    [ 29, 41, 1637, 2311, 3457 ],
    [ 7, 13, 83 ],
    [ 41, 61, 2383 ],
    [ 5, 17, 59 ],
    [ 3, 5, 7 ],
    [ 5, 107 ],
    [ 3, 23, 73 ],
    [ 3, 5, 67, 71 ],
    [ 499, 2203, 2503 ],
    [ 107, 269, 1367 ],
    [ 557, 727 ],
    [ 47, 241, 337, 727 ],
    [ 3, 11, 37, 107 ],
    [ 197, 269, 1481, 3557 ],
    [ 3, 7 ],
    [ 157, 179, 313, 1039 ],
    [ 23, 43, 4339, 8641 ],
    [ 11, 37, 193, 509, 853, 1747, 1823 ],
    [ 37, 89, 199, 661, 4127, 4217, 7549 ],
    [ 11, 23, 149 ],
    [ 5, 97, 307, 709, 1201, 4013 ],
    [ 19, 223, 631, 691, 1619 ],
    [ 97, 113, 163 ],
    [ 5, 11, 23 ],
    [ 19, 97 ],
    [ 5, 7, 23, 71 ],
    [ 13, 17, 23, 71, 79 ],
    [ 59, 73, 907, 1567, 4457 ],
    [ 19, 379, 383, 643 ],
    [ 3, 13, 83, 103, 269 ],
    [ 11 ],
    [ 97, 1249, 3109, 5807, 6863, 8737 ],
    [ 367, 523, 881, 1009 ],
    [ 103, 197 ],
    [ 443 ],
    [],
    [ 19, 29, 37, 263, 373, 911, 3313, 9551 ],
    [ 3, 37 ],
    [ 151, 239, 1549 ],
    [ 271 ],
    [ 47, 251, 641, 2011, 2969, 3539 ],
    [ 53, 487, 1753, 4373 ],
    [ 101, 2311 ],
    [ 43, 359, 491, 787, 1783 ],
    [ 13, 23, 41, 53, 233, 283, 571, 7853, 9721 ],
    [ 47, 97, 163, 263, 601, 1511, 1861, 1873, 2153, 3209, 7607 ],
    [ 17, 1181, 6967 ],
    [ 3, 23, 31, 61 ],
    [ 29, 89, 101, 367, 617, 1291, 1483, 7873, 8329 ],
    [ 19, 409 ],
    [ 11 ],
    [ 11, 23, 107 ],
    [],
    [ 3, 11, 23 ],
    [ 149 ],
    [ 23, 41, 71, 157, 263 ],
    [ 149, 5417 ],
    [ 541, 773 ],
    [ 19, 659, 853 ],
    [ 149, 2141, 2797, 3413, 3847 ],
    [ 3, 59, 83 ],
    [ 127, 157, 811, 1571, 2039, 6101 ],
    [ 421, 1319 ],
    [ 43, 53 ],
    [ 3, 97, 109 ],
    [ 13, 103 ],
    [ 3, 167 ],
    [ 3, 5 ],
    [ 101, 251 ],
    [ 3, 43, 263, 1447 ],
    [ 659, 1049, 2347, 6397 ],
    [ 3, 97, 181, 227 ],
    [ 29, 2287, 5749 ],
    [ 19, 167, 1613, 2903, 6709 ],
    [ 5 ],
    [ 83, 211, 383, 479, 2699 ],
    [ 5, 41, 109 ],
    [ 71, 79, 4943 ],
    [ 13 ],
    [ 449, 641, 661, 4261 ],
    [ 193, 199 ],
    [ 3, 5, 37 ],
    [ 467, 701, 7213 ],
    [ 433, 541, 547, 569 ],
    [ 19, 61, 149, 613, 1181 ],
    [ 3, 19, 47, 479, 619 ],
    [ 463, 487 ],
    [ 2099, 2161, 8537 ],
    [],
    [ 13, 191, 509, 2917 ],
    [ 97, 199, 647, 739 ],
    [],
    [ 521, 643 ],
    [ 83, 239, 2909, 3251, 4721 ],
    [ 5 ],
    [ 3, 5, 13, 23 ],
    [ 11, 43, 71 ],
    [ 103, 2663, 8719 ],
    [],
    [ 311, 557 ],
    [ 661, 881, 1097 ],
    [ 79 ],
    [ 11, 23, 101, 709, 719, 1453 ],
    [ 19 ],
    [ 3, 5, 109 ],
    [ 7, 31, 47 ],
    [ 31, 409 ],
    [ 7, 17, 19, 41, 233 ],
    [ 61, 137 ],
    [ 3, 5, 89, 103 ],
    [ 41, 53, 71, 449, 2677 ],
    [ 7, 19, 83, 593, 599 ],
    [ 7, 19, 43, 107, 2843, 6577, 6703 ],
    [ 3, 97, 307, 311 ],
    [ 127, 181 ],
    [ 3, 139, 157, 947 ],
    [ 11, 13 ],
    [ 23 ],
    [ 5, 197, 613, 5591, 6619 ],
    [ 3, 5, 43 ],
    [ 53 ],
    [ 11, 239, 313, 379 ],
    [ 89, 1279, 1303 ],
    [ 307, 1999 ],
    [ 3, 7 ],
    [ 5 ],
    [],
    [ 103, 139, 241, 587 ],
    [ 23, 251, 593, 613 ],
    [ 467 ],
    [ 41, 53 ],
    [ 379, 409, 491 ],
    [ 137, 179, 211, 541 ],
    [ 3, 5, 59, 167 ],
    [ 7, 179, 193, 1237, 6781 ],
    [ 43, 677, 761, 1597, 5653 ],
    [ 61, 67, 487, 521, 3673 ],
    [ 7 ],
    [ 43, 1481 ],
    [ 23, 167, 181, 331, 661, 3943 ],
    [ 79, 97, 103 ],
    [ 3 ],
    [ 11 ],
    [ 13, 17, 23 ],
    [ 3 ],
    [],
    [ 313, 467, 569, 3607 ],
    [ 13, 43, 1171, 1231 ]
];

//#[i : i in [1..#necessary_MWS_primes] |necessary_MWS_primes[i] eq [] ]; //8 = 6+2