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R-Simplex's the number 5 and prime numbers.

  1. Mar 14, 2015 #1
    So if you do a search for R-Simplexs you should find that.

    RSimplex(n,d)=Pochhammer(n,d)/d!

    Well so to does
    RSimplex(n,d)=If(n<d, Pochhammer(d+1,n-1)/n!, Pochhammer(n,d)/d!)

    Or something like that my maths package is down so I'm not sure quite how it works.

    Anyway the relationship between RSimplex's and primes seems to be this.

    For any tested prime over 11
    PrimeQ(p)=IntegerQ(Floor(RSimplex(RSimplex(p,p),5)/p)/p)
     
  2. jcsd
  3. Mar 14, 2015 #2

    Dale

    Staff: Mentor

    Please provide more information. I did a search and did not find that. What language/library/etc. are you talking about? What is your source?
     
  4. Mar 15, 2015 #3
    Pochhammer is the same as the rising factorial.
    I use mathematica but I wrote this article in pseudo code.
    Sorry about the search issue. Search instead for Figurate number, Tetrahedral number and Pochhammer.
    It all used to be on one page on the wikipedia now the validating info is scattered. I don't know why.
     
  5. Mar 15, 2015 #4

    Dale

    Staff: Mentor

    You should test non-primes also. For example, 195 is not prime, but the expression on the right evaluates to True as far as I can tell.
     
  6. Mar 15, 2015 #5
    Thats interesting my computer is down so I can't yet validate if your findings are the same as mine. I will get back to you as soon as and tell you if I get the same as yours. Just out of curiosity has your anaysis flagged any numbers that are not a multiple of 5 claiming true especially ones made up larger factors than 11.
     
  7. Apr 4, 2015 #6
    Yes I have now been able to do a bit of testing on my function and other than numbers below 11 or
    divisible by five I can't find any more which don't work.

    I Also found this test which Quite possibly can be sped up at least it's one step better than the last test in terms of speed up.

    PrimeQu[a34_] :=
    If[Mod[FromDigits[Append[Table[1, {xr, a34 - 1}], 0], 2], a34] == 0,
    If[Mod[FromDigits[Append[Table[1, {xd, a34 - 1}], 0], 5], a34] ==
    0 || a34 == 2, True, False], False];

    Essentially for every number I have n-1 1's in a list with a 0 at the end and you convert from binary and base 5 and then mod the output by n if both the numbers are 0 then it seems as if the number is a prime.
    Note this might only be a probable prime test who knows but it is displaying some very promising results.

    If you have any thoughts on how this can be speed up please let me know. I noticed how other primeality tests rely on modulation of big powers I see this technique as being similar so I'm thinking there's probably a bitwise trick out there but am I right. I'm just left killing the ram otherwise If I go beyond a billion as I have to then compute a billion 1's or so. I would love to know the score with regards to this issue.
     
  8. Apr 4, 2015 #7
    I've a made some Improvements already and yet again 5 and this time 2 and 3 also play a major role.I'm currently running 3000000 random number test's on the function so far it has thrown up no false or true lies compared to the standard wolfram test.

    PrimeQu[cex_] := Catch[ Module[ {zcv = PowerMod[2, cex, cex] - 2, zcv2 = JacobiSymbol[ FromDigits[Append[Table[1, {xvc, Floor[Log[2, cex]] + 1}], 0], 3] FromDigits[ Append[Table[1, {xvc, Floor[Log[3, cex]] + 1}], 0], 2], cex], zcv3 = JacobiSymbol[ FromDigits[Append[Table[1, {xvc, Floor[Log[2, cex]] + 1}], 0], 5] FromDigits[ Append[Table[1, {xvc, Floor[Log[5, cex]] + 1}], 0], 2], cex], Var25, Var26, ced = cex }, Var25 = If[GCD[ced, 2] == 2 || GCD[ced, 3] == 3 || GCD[ced, 5] == 5, If[ced == 2 || ced == 3 || ced == 5, 1, 2], 1]; Var26 = If[zcv == 0 && Abs[zcv2] == 1 && Abs[zcv3] == 1, 1, 2]; If[Var25 == 1 && Var26 == 1, Throw["True"], Throw["False"]]; ]];
     
    Last edited: Apr 5, 2015
  9. Apr 7, 2015 #8
    Check this code out this should get a few more views I've been beavering away at this one and am currently working on Optimised version. Any faulty numbers please do tell especially ones with similar stats to 521.

    Ok here goes it.

    Test1[vbe_] :=
    Catch[Module[{ce1 = vbe, zcv2, zcv3, zcv4, zcv5, zcv6, zcv7, zcv8,
    zcv9, zcv10, zcv11},
    zcv2 =
    JacobiSymbol[ce1,
    FromDigits[
    Append[Table[1, {xvc1 , Round[N[Log[2, ce1], 3]]}], 0],
    3] FromDigits[
    Append[Table[1, {xvc2, Round[N[Log[3, ce1], 3]]}], 0], 2]];
    zcv3 =
    JacobiSymbol[ce1,
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[2, ce1], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce1], 3]]}], 0], 2]];
    zcv4 =
    JacobiSymbol[ce1,
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[3, ce1], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce1], 3]]}], 0], 3]];
    Throw[{zcv2, zcv3, zcv4}]
    ]];

    Test2[vbt_] := Catch[Module[{ce2 = vbt, zcv5, zcv6, zcv7},
    zcv5 =
    JacobiSymbol[ce2 2,
    FromDigits[
    Append[Table[1, {xvc1, Round[N[Log[2, ce2], 3]]}], 0],
    3] FromDigits[
    Append[Table[1, {xvc2, Round[N[Log[3, ce2], 3]]}], 0], 2]];
    zcv6 =
    JacobiSymbol[ce2 2,
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[2, ce2], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce2], 3]]}], 0], 2]];
    zcv7 =
    JacobiSymbol[ce2 2,
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[3, ce2], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce2], 3]]}], 0], 3]];

    Throw[{zcv5, zcv6, zcv7}]
    ]];

    Test3[vbs_] := Catch[Module[{ce3 = vbs, zcv8, zcv9, zcv10},
    zcv8 =
    JacobiSymbol[ce3^2,
    FromDigits[Append[Table[1, {xvc1, Floor[Log[2, ce3]] + 1}], 0],
    3] FromDigits[Append[Table[1, {xvc2, Floor[Log[3, ce3]]}], 0],
    2]];
    zcv9 =
    JacobiSymbol[ce3^2,
    FromDigits[Append[Table[1, {xvc3, Floor[Log[2, ce3]] + 1}], 0],
    5] FromDigits[Append[Table[1, {xvc4, Floor[Log[5, ce3]]}], 0],
    2]];
    zcv10 =
    JacobiSymbol[ce3^2,
    FromDigits[Append[Table[1, {xvc3, Floor[Log[3, ce3]] + 1}], 0],
    5] FromDigits[Append[Table[1, {xvc4, Floor[Log[5, ce3]]}], 0],
    3]];

    Throw[{zcv8, zcv9, zcv10}]
    ]];
    Test6[vbe_] :=
    Catch[Module[{ce1 = vbe, zcv2, zcv3, zcv4, zcv5, zcv6, zcv7, zcv8,
    zcv9, zcv10, zcv11},
    zcv2 =
    JacobiSymbol[
    FromDigits[
    Append[Table[1, {xvc1 , Round[N[Log[2, ce1], 3]]}], 0],
    3] FromDigits[
    Append[Table[1, {xvc2, Round[N[Log[3, ce1], 3]]}], 0], 2],
    ce1];
    zcv3 =
    JacobiSymbol[
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[2, ce1], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce1], 3]]}], 0], 2],
    ce1];
    zcv4 =
    JacobiSymbol[
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[3, ce1], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce1], 3]]}], 0], 3],
    ce1];
    Throw[{zcv2, zcv3, zcv4}]
    ]];

    Test7[vbt_] := Catch[Module[{ce2 = vbt, zcv5, zcv6, zcv7},
    zcv5 =
    JacobiSymbol[
    FromDigits[
    Append[Table[1, {xvc1, Round[N[Log[2, ce2], 3]]}], 0],
    3] FromDigits[
    Append[Table[1, {xvc2, Round[N[Log[3, ce2], 3]]}], 0], 2],
    ce2 5];
    zcv6 =
    JacobiSymbol[
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[2, ce2], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce2], 3]]}], 0], 2],
    ce2 5];
    zcv7 =
    JacobiSymbol[
    FromDigits[
    Append[Table[1, {xvc3, Round[N[Log[3, ce2], 3]]}], 0],
    5] FromDigits[
    Append[Table[1, {xvc4, Round[N[Log[5, ce2], 3]]}], 0], 3],
    ce2 5];

    Throw[{zcv5, zcv6, zcv7}]
    ]];

    Test8[vbs_] := Catch[Module[{ce3 = vbs, zcv8, zcv9, zcv10},
    zcv8 =
    JacobiSymbol[
    FromDigits[Append[Table[1, {xvc1, Floor[Log[2, ce3]] + 1}], 0],
    3] FromDigits[Append[Table[1, {xvc2, Floor[Log[3, ce3]]}], 0],
    2], ce3^3];
    zcv9 =
    JacobiSymbol[
    FromDigits[Append[Table[1, {xvc3, Floor[Log[2, ce3]] + 1}], 0],
    5] FromDigits[Append[Table[1, {xvc4, Floor[Log[5, ce3]]}], 0],
    2], ce3^3];
    zcv10 =
    JacobiSymbol[
    FromDigits[Append[Table[1, {xvc3, Floor[Log[3, ce3]] + 1}], 0],
    5] FromDigits[Append[Table[1, {xvc4, Floor[Log[5, ce3]]}], 0],
    3], ce3^3];

    Throw[{zcv8, zcv9, zcv10}]
    ]];

    Test4[vbr_] := Catch[Module[{zcv, zcv11, fbe = vbr},
    zcv = PowerMod[2, fbe, fbe] - 2;
    zcv11 = PowerMod[3, fbe, fbe] - 3;
    Throw[{zcv, zcv11}]
    ]];

    Test5[vbr2_] :=
    Catch[Module[{Vbry = vbr2,
    asnd = Append[Table[1, {ghd, vbr2 - 1}], 0], tes1, tes2},
    tes1 = Mod[FromDigits[asnd, 2], Vbry];
    tes2 = Mod[FromDigits[asnd, 5], Vbry];
    Throw[{tes1, tes2}]
    ]];


    PrimeQPW[cex_] :=
    Catch[Module[{Var27, Var28, Var29, Var30, Var26 = Test1[cex],
    Var25 = Abs[Test2[cex]], Var24 = Test3[cex], Var23 = Test4[cex],
    ced = cex},
    If[ced == 2 || ced == 3 || ced == 5 || ced == 7 || ced == 11 ||
    ced == 13, Throw[True]; Break[];];
    If[IntegerQ[ced/2] == True || IntegerQ[ced/3] == True ||
    IntegerQ[ced/5] == True || IntegerQ[ced/7] == True ||
    IntegerQ[ced/11] == True || IntegerQ[ced/13] == True,
    Throw[False]; Break[];];
    If[Total[Abs[Var26]] < 2, Throw[False]; Break[], Var27 = 1];
    If[Total[Var25] > 1, Throw[False]; Break[], Var28 = 1];
    If[Var26[[2]] == -1 && Total[Var24] < 3 || Total[Var24] == 3,
    Var29 = 1;, Throw[False]; Break[];];
    Var30 = If[Var23 == {0, 0}, 1, 2];
    If[Var27 == 1 && Var28 == 1 && Var29 == 1 && Var30 == 1,
    Throw[True];, Throw[False];];
    ]];


    Stats[sta_] :=
    Column[{{Test1[sta], Test2[sta], Test3[sta], Test6[sta], Test7[sta],
    Test8[sta]},
    FromDigits[{Mod[sta, 13], Mod[sta, 11], Mod[sta, 7], Mod[sta, 5]},
    13]}];

    It seems to work fastest with Big Primes Over 500 digits long. enjoy. And Again any faulty results do post by all means especially ones with interesting stats or ones faster or slower than the norm with regards to big Primes.
     
  10. Apr 7, 2015 #9
    488881 gives a false reading.
     
  11. Apr 9, 2015 #10
    Watch the Zero count on this test in relation to groups of 1's and -1's for primes and non primes. I'm about to write these findings into a test.

    FiveItt[x98_, cc5_] :=
    If[x98 == 1, 1, FromDigits[Append[Table[1, {ft, x98 - 1}], 0], cc5]];

    BCC[x55_, g77_] :=
    Drop[Flatten[
    Reap[Module[{x45 = x55, z7 = 0, z8 = 0, z9, g7 = g77, bell},
    z7 = If[x45/FiveItt[Length[IntegerDigits[x45, g7]], g7] <= 1,
    If[x45 == 1, 1, Length[IntegerDigits[x45, g7]] - 1],
    Length[IntegerDigits[x45, g7]]];
    bell = FiveItt[z7 - 1, g7];
    z9 = g7^(z7 - 1);
    Label[SPo];
    z8 =
    If[IntegerQ[x45/g7] && x45 > g7,
    Quotient[x45 - bell - (1/(2*g7)), z9],
    If[x45 <= g7, x45, Quotient[x45 - bell, z9]]];
    Sow[z8];
    x45 = x45 - (z8*(z9));
    z7 = z7 - 1;
    z9 = z9/g7;
    bell = bell - z9;
    If[z7 < 1, Goto[EnD], Goto[SPo]];
    Label[EnD];]]], 1];

    T2xy[x_, y_] := ((x + y) (x + y - 1))/2 - y + 1;
    Zequeba[fre_, fre2_] := Total[BCC[fre, fre2]];

    Test10[zz7_,
    zz8_] := {JacobiSymbol[Zequeba[zz7, zz8], T2xy[zz7, zz8]],
    JacobiSymbol[T2xy[zz7, zz8], Zequeba[zz7, zz8]],
    JacobiSymbol[Zequeba[zz7, zz8], T2xy[zz8, zz7]],
    JacobiSymbol[T2xy[zz8, zz7], Zequeba[zz7, zz8]]};

    Lightening[asif_] := Table[Test10[asif, Prime[hha]], {hha, 1, 6}];
     
  12. Apr 14, 2015 #11
    This testing Mechanism has a very low big O Optimisability and Could Radically speed up Prime Search's. It also Correlates well with SHA Hashs 256 368 and 512.

    Test4[vbr_] := Catch[Module[{zcv, zcv11, fbe = vbr},
    zcv = PowerMod[2, fbe, fbe] - 2;
    zcv11 = PowerMod[3, fbe, fbe] - 3;
    Throw[{zcv, zcv11}]
    ]];

    FiveItt[x98_, cc5_] :=
    If[x98 == 1, 1,
    FromDigits[Append[Table[1, {ft, x98 - 1}], 0], cc5]];

    BCCLog[x22_, y22_] :=
    Catch[Module[{s21 = x22, t22 = Length[IntegerDigits[x22, y22]]},
    If[FiveItt[t22, y22] > s21, t22 = t22 - 1];
    Throw[t22];
    ]];
    T2xy[x_, y_] := ((x + y) (x + y - 1))/2 - y + 1;

    Fifty[tas_, ras_] :=
    Catch[Module[{tar = tas, rar = ras, leng1, fvar1, fvar2, fvar3},
    fvar1 = Prime[rar];
    fvar2 = Length[IntegerDigits[tar, fvar1]];
    fvar3 = FiveItt[fvar2, fvar1];
    If[fvar3 > tar, fvar2 = fvar2 - 1];
    fvar3 = IntegerQ[Mod[(FiveItt[fvar2, fvar1] + tar), tar]/fvar2];
    Throw[fvar3];
    ]];

    Test10[aa7_, aa8_] :=
    Catch[Module[{zz7 = aa7, zz8 = aa8, zz9 = T2xy[aa8, aa7],
    zz10 = T2xy[aa7, aa8]},
    Throw[{JacobiSymbol[zz7, zz10], JacobiSymbol[zz7, zz9],
    JacobiSymbol[zz10, zz7], JacobiSymbol[zz9, zz7]}]]];

    Test11[r56_, r58_] :=
    Catch[Module[{x57 = r56, x58 = r58, x51, x52, d43, d44},
    d44 = Prime[x58];
    x51 = Length[IntegerDigits[x57, d44]];
    x52 = FiveItt[x51, d44];
    If[x52 > x57, x51 = x51 - 1];
    d43 =
    Flatten[Table[Test10[x57, r57*x51], {r57, x57, x57 + d44}]];

    Throw[d43];
    ]];

    MiniTest[y44_] :=
    Catch[Module[{y45 = y44, y67 = T2xy[y44 + 5, y44],
    y68 = T2xy[y44, y44 + 5]},
    Throw[{JacobiSymbol[y67, y45], JacobiSymbol[y68, y45],
    JacobiSymbol[y45, y67], JacobiSymbol[y45, y68]}]]];

    Test14[t13_, t16_] := Catch[Module[{t14 = t13, Data12, t15 = t16},
    Data12 = Test11[t14, t15];
    Total[{Count[Data12, 1], Count[Data12, -1]}]

    ]];

    PrimeQPW[hh3_] := Catch[Module[
    {aa3, aa4 = hh3, aa5, aa6, aa7, aa8, aa9, a10, a11, a12, a13, a14,
    a15, a16, aam},
    aa5 = 1;

    Label["Whittle"];
    If[aa4 == Prime[aa5] || aa4 == Prime[aa5 + 1] ||
    aa4 == Prime[aa5 + 2], Throw[True]; Break[];];
    If[aa5 >= 25, aa5 = 1; Goto["Whittle2"] ;, aa5 = aa5 + 3;
    Goto["Whittle"];];

    Label["Whittle2"];
    If[IntegerQ[aa4/Prime[aa5]] == True ||
    IntegerQ[aa4/Prime[aa5 + 1]] == True ||
    IntegerQ[aa4/Prime[aa5 + 2]] == True, Throw[False]; Break[];];
    If[aa5 >= 25, aa5 = 1; Goto["NextStep"] ;, aa5 = aa5 + 3;
    Goto["Whittle2"];];

    Label["NextStep"];
    If[aa4 < Prime[25]^2, Throw[True]; Break[];];
    aa8 = Test14[aa4^2, 5];
    If[aa8 == 48, aa6 = 1; aa3 = {1};];
    If[aa8 == 46, aa3 = {1}; aa6 = 2;];
    If[aa6 == 1 || aa6 == 2, Goto["NextStep2"];];
    Goto["FalseEnd"];

    Label["NextStep2"];
    aa3 = Append[aa3, Test14[aa4*Prime[aa5], 5]];
    aa5++;
    If[aa5 > 5, aa5 = 1;
    If[aa6 == 1, Goto["CheckRecords"];, Goto["CheckRecords2"];];,
    Goto["NextStep2"];];

    Label["CheckRecords"];
    aa3 = Drop[aa3, 1];
    If[aa3 == {0, 32, 40, 36, 40} , aa6 = 0; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 38, 40, 40}, aa6 = 1; Goto["CheckPoint"]; ];
    If[aa3 == {24, 32, 40, 36, 40}, aa6 = 2; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 40, 36, 40}, aa6 = 3; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 40, 40, 40}, aa6 = 4; Goto["CheckPoint"];];
    If[aa3 == {24, 46, 38, 38, 38}, aa6 = 5; Goto["CheckPoint"];];
    If[aa3 == {22, 46, 38, 36, 40}, aa6 = 6; Goto["CheckPoint"];];
    If[aa3 == {22, 46, 38, 34, 38}, aa6 = 7; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 38, 34, 38}, aa6 = 8; Goto["CheckPoint"];];
    If[aa3 == {0, 30, 38, 34, 38}, aa6 = 9; Goto["CheckPoint"];];
    If[aa3 == {0, 32, 40, 40, 40}, aa6 = 10; Goto["CheckPoint"];];
    If[aa3 == {0, 32, 48, 40, 40}, aa6 = 11; Goto["CheckPoint"];];
    If[aa3 == {24, 32, 48, 40, 40}, aa6 = 12; Goto["CheckPoint"];];
    If[aa3 == {24, 32, 40, 40, 40}, aa6 = 13; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 38, 36, 48}, aa6 = 20; Goto["CheckPoint"];];
    If[aa3 == {0, 48, 38, 36, 40}, aa6 = 21; Goto["CheckPoint"];];
    If[aa3 == {24, 32, 38, 40, 40}, aa6 = 22; Goto["CheckPoint"];];
    If[aa3 == {0, 32, 38, 40, 40}, aa6 = 23; Goto["CheckPoint"];];
    If[aa3 == {24, 32, 38, 36, 40}, aa6 = 24; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 48, 36, 40}, aa6 = 25; Goto["CheckPoint"];];
    If[aa3 == {24, 32, 48, 48, 40}, aa6 = 26; Goto["CheckPoint"];];
    If[aa3 == {24, 32, 40, 48, 40}, aa6 = 27; Goto["CheckPoint"];];
    If[aa3 == {0, 32, 40, 48, 40}, aa6 = 28; Goto["CheckPoint"];];
    If[aa3 == {0, 48, 38, 40, 40}, aa6 = 29; Goto["CheckPoint"];];
    If[aa3 == {24, 32, 38, 48, 40}, aa6 = 30; Goto["CheckPoint"];];

    Goto["FalseEnd"];

    Label["CheckRecords2"];
    aa3 = Drop[aa3, 1];
    If[aa3 == {24, 48, 40, 40, 38}, aa6 = 14; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 40, 36, 40}, aa6 = 15; Goto["CheckPoint"];];
    If[aa3 == {24, 48, 40, 40, 40}, aa6 = 16; Goto["CheckPoint"];];
    If[aa3 == {24, 30, 38, 34, 38}, aa6 = 17; Goto["CheckPoint"];];
    If[aa3 == {0, 32, 40, 36, 40}, aa6 = 18; Goto["CheckPoint"];];
    If[aa3 == {0, 32, 40, 40, 40}, aa6 = 19; Goto["CheckPoint"];];
    Goto["FalseEnd"];

    Label["CheckPoint"];
    aa5 = {{Mod[aa4, 3], Mod[aa4, 5]}, {Mod[aa4, 7], Mod[aa4, 11]}};
    aa7 = MiniTest[aa4];
    aa9 = {Fifty[aa4, 1], Fifty[aa4, 2], Fifty[aa4, 3], Fifty[aa4, 4],
    Fifty[aa4, 5]};
    a10 = Floor[Total[aa3]/5];
    a11 = GatherBy[Flatten[aa5], OddQ];
    If[Length[a11] == 1,
    If[AnyTrue[a11[[1]], OddQ] == True, a12 = {4, 0};,
    a12 = {0, 4};],
    If[AnyTrue[a11[[1]], OddQ] == True,
    a12 = {Length[a11[[1]]], Length[a11[[2]]]};,
    a12 = {Length[a11[[2]]], Length[a11[[1]]]};];];
    a11 = {GatherBy[aa5[[1]], OddQ], GatherBy[aa5[[2]], OddQ]};
    If[Length[a11[[1]]] == 1,
    If[EvenQ[Flatten[a11[[1]]][[1]]] == True, a13 = {0, 2};,
    a13 = {2, 0};];, a13 = {1, 1};];
    If[Length[a11[[2]]] == 1,
    If[EvenQ[Flatten[a11[[2]]][[1]]] == True, a14 = {0, 2};,
    a14 = {2, 0};];, a14 = {1, 1};];
    a11 = {a13, a14};
    a12 = Append[{a12}, {a13, a14}];
    a11 = GatherBy[Flatten[aa5], PrimeQ];
    If[Length[a11] == 1,
    If[AnyTrue[a11[[1]], PrimeQ] == True, a13 = 4, a13 = 0];,
    If[AnyTrue[a11[[1]], PrimeQ] == True, a13 = Length[a11[[1]]];,
    a13 = Length[a11[[2]]]];];
    a14 = Count[Flatten[aa5], 1];
    a15 = Count[Flatten[aa5], 2];
    a12 = Append[a12, {a13, a14, a15}];
    aam = {0};
    If[aa7 == {1, 1, 1, 1}, Goto["Check1111"];];
    Goto["CheckPoint2a"];

    Label["CheckPoint2a"];
    Throw["2"]; Break[];

    Label["Check1111"];
    If[IntegerQ[Sqrt[aa4]] == True, Throw[False]; Break[];];
    If[aa9 == {True, False, False, False, False},
    Goto["1111abbbb"];];
    If[aa9 == {False, False, False, True, True}, Goto["1111bbbaa"];];
    If[aa9 == {False, False, True, False, True}, Goto["1111bbaba"];];
    If[aa9 == {False, True, True, False, True}, Goto["1111baaba"];];
    If[aa9 == {False, False, True, False, False},
    Goto["1111bbabb"];];
    If[aa9 == {False, True, False, False, True}, Goto["1111babba"];];
    If[aa9 == {False, True, False, True, True}, Goto["1111babaa"];];
    If[aa9 == {False, False, False, True, False},
    Goto["1111bbbab"];];
    If[aa9 == {True, False, False, True, False}, Goto["1111abbab"];];
    If[aa9 == {False, False, False, False, False}, Goto["1111bbbbb"];];
    If[aa9 == {False, True, False, False, False},
    Goto["1111babbb"];];
    If[aa9 == {True, False, False, False, True}, Goto["1111abbba"];];
    Goto["CheckPoint2a"];

    Label["1111abbbb"];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {4, 0, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{1, 3}, {{1, 1}, {0, 2}}, {1, 1, 1}},
    aam = Append[aam, 1]];
    Label["1111abbbbS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111bbbaa"];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {2, 1, 0}},
    aam = Append[aam, 2]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {4, 0, 2}},
    aam = Append[aam, 2]];
    If[a12 == {{1, 3}, {{1, 1}, {0, 2}}, {1, 1, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {2, 1, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{0, 4}, {{0, 2}, {0, 2}}, {1, 0, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {3, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {4, 0, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {3, 1, 3}},
    aam = Append[aam, 1]];
    If[a12 == {{4, 0}, {{2, 0}, {2, 0}}, {1, 2, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {1, 2, 0}},
    aam = Append[aam, 1]];
    Label["1111bbbaaS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111bbaba"];
    If[a12 == {{2, 2}, {{0, 2}, {2, 0}}, {4, 0, 2}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {2, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{0, 2}, {2, 0}}, {2, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {2, 1, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {2, 0, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {2, 1, 1}},
    aam = Append[aam, 2]];
    Label["1111bbabaS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111baaba"];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {2, 0, 2}},
    aam = Append[aam, 1]];
    Label["1111baabaS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111bbabb"];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {2, 2, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {1, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {3, 1, 2}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {2, 2, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {1, 2, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {3, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {1, 2, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{0, 2}, {2, 0}}, {3, 0, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {3, 0, 2}},
    aam = Append[aam, 1]];
    If[a12 == {{1, 3}, {{1, 1}, {0, 2}}, {1, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {1, 1, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {4, 0, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {1, 2, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {2, 1, 1}},
    aam = Append[aam, 1]];
    Label["1111bbabbS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111babba"];
    If[a12 == {{4, 0}, {{2, 0}, {2, 0}}, {1, 3, 0}},
    aam = Append[aam, 2]];
    If[a12 == {{2, 2}, {{2, 0}, {0, 2}}, {0, 2, 0}},
    aam = Append[aam, 2]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {3, 0, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {3, 1, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {2, 0, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{0, 4}, {{0, 2}, {0, 2}}, {2, 0, 2}},
    aam = Append[aam, 2]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {4, 0, 3}},
    aam = Append[aam, 2]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {1, 2, 0}},
    aam = Append[aam, 2]];
    If[a12 == {{4, 0}, {{2, 0}, {2, 0}}, {1, 2, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {2, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {2, 2, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{4, 0}, {{2, 0}, {2, 0}}, {2, 1, 0}},
    aam = Append[aam, 1]];
    Label["1111babbaS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111babaa"];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {1, 1, 0}},
    aam = Append[aam, 2]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {3, 0, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {2, 2, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{0, 2}, {2, 0}}, {4, 0, 2}},
    aam = Append[aam, 1]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {1, 1, 1}},
    aam = Append[aam, 1]];
    Label["1111babaaS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111bbbab"];
    If[a12 == {{0, 4}, {{0, 2}, {0, 2}}, {2, 0, 2}},
    aam = Append[aam, 2]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {1, 2, 0}},
    aam = Append[aam, 2]];
    Label["1111bbbabS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111abbab"];
    If[a12 == {{4, 0}, {{2, 0}, {2, 0}}, {1, 2, 0}},
    aam = Append[aam, 2]];
    Label["1111abbabS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111bbbbb"];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {2, 0, 1}},
    aam = Append[aam, 2]];
    If[a12 == {{2, 2}, {{0, 2}, {2, 0}}, {4, 0, 2}},
    aam = Append[aam, 1]];
    If[a12 == {{1, 3}, {{0, 2}, {1, 1}}, {1, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{2, 2}, {{1, 1}, {1, 1}}, {2, 1, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{0, 4}, {{0, 2}, {0, 2}}, {2, 0, 2}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {1, 2, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {2, 2, 1}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {3, 1, 1}},
    aam = Append[aam, 1]];
    Label["1111bbbbbS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111babbb"];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {2, 1, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {1, 2, 0}},
    aam = Append[aam, 1]];
    If[a12 == {{3, 1}, {{1, 1}, {2, 0}}, {3, 1, 1}},
    aam = Append[aam, 1]];
    Label["1111babbbS2"];
    Throw[{a12, aam}]; Break[];

    Label["1111abbba"];
    If[a12 == {{3, 1}, {{2, 0}, {1, 1}}, {2, 1, 0}},
    aam = Append[aam, 1]];
    Label["1111abbbaS2"];
    Throw[{a12, aam}]; Break[];

    Label["CheckIfTrue"];

    Label["FalseEnd"];
    Throw["1"];
    ]];

    Details[ww5_, ww6_] :=
    Column[{{Mod[ww5, 3], Mod[ww5, 5], Mod[ww5, 7],
    Mod[ww5, 11]}, {Test14[ww5^2, ww6], Test14[ww5 2, ww6],
    Test14[ww5 3, ww6], Test14[ww5 5, ww6], Test14[ww5 7, ww6],
    Test14[ww5 11, ww6]},
    MiniTest[ww5], {Fifty[ww5, 1], Fifty[ww5, 2], Fifty[ww5, 3],
    Fifty[ww5, 4], Fifty[ww5, 5]}}];

    Details[2^57885161 - 1, 5]
     
  13. Jun 1, 2015 #12
    I thought I would/should move to number maps for factorisation. Watch how much it dip's each Number compared to the previous number on the map.


    Px[x_] :=
    x - 1/2 Floor[
    1/2 (-1 + Sqrt[-7 + 8 x])] (1 + Floor[1/2 (-1 + Sqrt[-7 + 8 x])]);

    Py[x_] :=
    1/2 (2 - 2 x + Ceiling[1/2 (-1 + Sqrt[1 + 8 x])] +
    Ceiling[1/2 (-1 + Sqrt[1 + 8 x])]^2);

    Pxy[x_, y_] := ((x + y) (x + y - 1))/2 - y + 1;

    EItt[x2a_, y2a_, z2a_] :=
    FromDigits[
    Flatten[Append[Table[1, {t, y2a - z2a}], Table[0, {t2, z2a}]]],
    x2a];

    {BCCLog Instead of ELog}

    mod311[cu_] := Catch[Module[{cz = cu, z1, zz, zz1},
    zz = Px[cz];
    zz1 = Py[cz];
    If[OddQ[zz] == True && EvenQ[zz1] == True ||
    OddQ[zz] == True && EvenQ[zz1] == True, z1 = zz + zz1;];
    If[OddQ[zz] == True && OddQ[zz1] == True, z1 = zz*zz1;,
    z1 = zz*zz1 - 1;];
    Throw[z1];
    ]];
    Quota[k2_] :=
    Catch[Module[{k22 = k2, esta, esta2, esta3, esta4, esta5, esta5b,
    trip, tt, eset = 0},
    esta2 = Floor[Sqrt[k22]];
    trip = 0;
    esta3 = 1;
    tt = 2;

    Label["andbang"];
    esta =
    Quotient[
    Pxy[k22,
    Px[k22] + Py[k22]], (3 + Mod[k22, 2] + Mod[k22, 3]) BCCLog[k22,
    tt]];
    If[OddQ[esta] == True, esta = esta + 1];
    Goto["Start"];

    Label["Start"];
    If[IntegerQ[esta/3], eset = 1;, eset = 2;];
    esta =
    esta/2^IntegerExponent[esta, 2]/3^IntegerExponent[esta, 3]/
    5^IntegerExponent[esta, 5];
    If[esta <= esta2, Goto["Stage2"];,
    If[eset == 1, esta = esta - (mod311[esta] - 16);,
    esta = esta - (mod311[esta] - 2);]; Goto["Start"];];

    Label["Stage2"];
    If[PrimeQ[esta] == True, trip = (E (trip + E^2))];
    If[esta <= 11, Goto["Stage3"];];
    If[Length[esta3] != 0,
    esta3 = Append[esta3, {esta, Floor[trip/ 5]}];
    If[eset == 1, esta = esta - (mod311[esta] - 16);,
    esta = esta - (mod311[esta] - 2);]; Goto["Start"];,
    esta3 = {esta , Floor[trip/5]};];
    If[eset == 1, esta = esta - (mod311[esta] - 16);,
    esta = esta - (mod311[esta] - 2);]; Goto["Start"];
    Goto["End2"];

    Label["Stage3"];
    If[tt >= 5, esta4 = Append[esta4, esta3]; Goto["End1"];, trip = 0;
    If[tt == 2, tt = 3;, tt = NextPrime[tt + 1];];
    If[tt == 3, esta4 = {esta3};, esta4 = Append[esta4, esta3];];
    esta3 = 1; Goto["andbang"];];

    Label["End1"];
    esta4 = Floor[esta4];
    Throw[esta4];
    Break[];

    Label["End2"];
    Throw[{"Don't Know Why", esta3, esta}];
    ]];
    Stats[zxc_] := {Quota[zxc], Quota[zxc 2], Quota[zxc 3]};

    SeeMap[tt15_] :=
    Catch[Module[{tm4 = tt15, ratta, ratta2 = 1, ratta3 = 1, god = 1,
    raster = 1, devil = 1, ttin = 0, ttin2 = 0, ttin3, ttin4},
    ratta = Sort[DeleteDuplicates[Flatten[Stats[tm4]]]];
    ttin3 = E*1.06;
    ttin4 = Length[ratta];
    ratta3 = GatherBy[ratta, OddQ];
    If[PrimeQ[ratta3[[1]][[1]]], ratta3 = ratta3[[1]],
    If[Length[ratta3] >= 2, ratta3 = ratta3[[2]], ratta3 = {0}];];
    raster = Floor[Sqrt[tm4]];
    god = Quantile[ratta, 3/4];
    devil = Quantile[ratta, 1/2];
    Label["Bottom"];
    ttin++;
    ratta2 = Quantile[ratta, 1/4];
    If[ttin >= ttin4, Goto["fst"];, Goto["Bottom"]];
    Label["fst"];
    Throw[
    Throw[{{god, devil, ratta2, ttin4, raster}, {god - raster,
    Floor[(god)/(ratta2 + 1)], devil ttin4}, ratta3}]]

    ]];




    seeMap[NextPrime[30^100] NextPrime[30^101]]
     
    Last edited: Jun 1, 2015
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