1 | // $Id: groups.lib,v 1.3 2005-05-06 14:38:34 hannes Exp $ |
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2 | //(GP, last modified 26.10.01) |
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3 | /////////////////////////////////////////////////////////////////////////////// |
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4 | version="$Id: groups.lib,v 1.3 2005-05-06 14:38:34 hannes Exp $"; |
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5 | category="Applications"; |
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6 | info=" |
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7 | LIBRARY: groups.lib Finite Group Theory |
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8 | AUTHORS: Gert-Martin Greuel, email: greuel@mathematik.uni-kl.de @* |
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9 | Gerhard Pfister, email: pfister@mathematik.uni-kl.de |
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10 | |
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11 | PROCEDURES: |
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12 | |
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13 | noSolution(I) I an ideal in a polynomial ring over Z[x1..xn] |
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14 | returns a list l of primes <=32003 such that 1 |
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15 | is in IZ/p[x1..xn] for p not in l or an ERROR |
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16 | if this is wrong or not decided. |
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17 | " |
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18 | /* |
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19 | // ===================== A Problem in Finite Group Theory =================== |
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20 | // Posed by Boris Kunyavskii, Bar-Ilan University, Tel Aviv |
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21 | // |
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22 | // For any word w in X,Y,X^(-1),Y^(-1) consider the sequence U_n |
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23 | // of words (depending on w) inductively |
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24 | // U_1 = w |
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25 | // U_n+1 = [X*U_n*X^(-1),Y*U_n*Y^(-1)] |
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26 | // with |
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27 | // [X,Y] = X*Y*X^(-1)*Y^(-1) (the commutator) |
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28 | |
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29 | |
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30 | // Conjecture (1): |
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31 | // (by B. Plotkin, slightly modified): |
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32 | // A finite group G is solvable <==> |
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33 | // there is an n >= 1 such that U_n(x,y) = 1 for all x,y in G |
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34 | // and for any of the 4 following words |
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35 | // w1 = X^(-1)*Y*X*Y^(-1)*X |
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36 | // w2 = X^(-2)*Y^(-1)*X |
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37 | // w3 = Y^(-2)*X^(-1)*X |
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38 | // w4 = X*Y^(-2)*X^(-1)*Y*X^(-1) |
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39 | // |
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40 | // (These words remained as possibly good words by a computer search |
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41 | // through about 10 000 candidates, by Fritz Grunewald) |
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42 | |
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43 | |
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44 | // ==> is clear |
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45 | // The minimal finite non-solvable groups (i.e. every proper subgroup is |
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46 | // solvable) have been classified by Thompson in 1968, they are: |
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47 | // |
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48 | // 1. PSL(2,p) (p=5 or p=+-2 (mod 5), p!=3) |
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49 | // 2. PSL(2,2^p) |
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50 | // 3. PSL(2,3^p) (p odd) |
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51 | // 4. Sz(2^p) (p odd) (Suzuki group) |
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52 | // 5. PSL(3,3) |
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53 | // |
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54 | // In view of this result Conjecture (1) is equivalent to |
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55 | // |
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56 | // Conjecture (2): |
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57 | // Let G be one of the groups above, then, for at least one of the 4 words w |
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58 | // above there are x,y in G such that 1 != U_n(x,y) = U_n+1(x,y) for some n. |
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59 | // (then U_n(x,y) != 1 for all n by definitionof U_n) |
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60 | |
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61 | |
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62 | // We give a computer aided proof, using SINGULAR, of Conjecture (2) for |
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63 | // the groups 1. - 3. (up to checking for a small, explicit number of primes) |
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64 | // Hence only the Suzuki groups have to be checked. |
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65 | // |
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66 | // We show: 1 != U_1(x,y) = U_2(x,y) for word w1 and some x,y in G |
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67 | // We need: |
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68 | // Theory |
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69 | // - simple facts from algebraic geometry, singularity theory, finite fields |
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70 | // - theorem of Lang-Weil, estimating the number of rational points on an |
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71 | // absolutely irreducible projective curve C defined over Z (g=genus): |
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72 | // if q >= N(g) ==> #C(F_q) != 0 |
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73 | // (N(g) = 4g^2 -2, g arithmetic genus) |
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74 | // SINGULAR |
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75 | // - Groebner basis (elimination), multivariate factorization, resolution of |
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76 | // plane curve singularities (Hamburger Noether developement), primary |
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77 | // decomposition |
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78 | |
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79 | */ |
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80 | |
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81 | LIB "standard.lib"; |
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82 | LIB "general.lib"; |
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83 | LIB "matrix.lib"; |
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84 | /////////////////////////////////////////////////////////////////////////////// |
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85 | |
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86 | static proc splitS1(ideal I,int s,list #) |
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87 | "USAGE: splitS1(I,s[,l]) I ideal, s integer, l list of ideals |
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88 | COMPUTE: Factorizes the generators of I. If for one generator f=f1*f2 and |
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89 | fi=ni*gi^ri, ni an integer and gi a polynomial, then |
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90 | compute a standardbasis of I1=I,g1 and I2=I,g2, if this can be |
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91 | done in < s seconds. Then apply splitS to I1 and I2 and continue |
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92 | in the same way. The procedure stops if no generator can be |
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93 | factorized. |
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94 | RETURN: A list L of ideals and prime numbers |
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95 | L[1]: A list of ideals such that the radical of the intersection |
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96 | of these ideals coincides with the radical of I |
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97 | If the optional list l of the input is not empty then |
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98 | the ideals of L[1] which contain an ideal of l are canceled. |
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99 | L[2]: A list of prime numbers appearing as factors of the ni. |
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100 | NOTE: The computation avoids division by integers (by using |
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101 | option(contentSB)) hence the result is correct modulo any prime |
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102 | number which does not appear in the list L[2]. |
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103 | EXAMPLE: example splitS1; shows an example |
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104 | " |
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105 | { |
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106 | option(redSB); |
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107 | option(contentSB); |
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108 | int j,k,e; |
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109 | int i=1; |
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110 | int l=attrib(I,"isSB"); |
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111 | ideal J; |
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112 | list re,fac,te,pr,qr,w; |
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113 | number n; |
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114 | poly p; |
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115 | re=#; |
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116 | |
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117 | if(deg(I[1])==0){return(list(re+list(std(I)),qr));} |
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118 | |
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119 | fac=factorize(I[1]); |
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120 | |
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121 | while((size(fac[1])==2)&&(i<size(I))) |
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122 | { |
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123 | I[i]=fac[1][2]*fac[1][1]; //not in splitS |
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124 | i++; |
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125 | fac=factorize(I[i]); |
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126 | } |
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127 | if(size(fac[1])==2){I[size(I)]=fac[1][2]*fac[1][1];} //not in splitS |
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128 | if(size(fac[1])>2) |
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129 | { |
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130 | w=squarefreeP(number(fac[1][1])); |
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131 | n=w[1]; |
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132 | qr=insResult(qr,w[2]); |
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133 | for(j=2;j<=size(fac[1]);j++) |
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134 | { |
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135 | I[i]=fac[1][j]; |
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136 | attrib(I,"isSB",1); |
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137 | e=1; |
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138 | k=0; |
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139 | while(k<size(re)) |
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140 | { |
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141 | k++; |
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142 | if(size(reduce(re[k],I))==0){e=0;break;} |
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143 | attrib(re[k],"isSB",1); |
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144 | if(size(reduce(I,re[k]))==0){re=delete(re,k);k--;} |
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145 | } |
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146 | if(j==2){I[i]=I[i]*n;} |
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147 | if(e) |
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148 | { |
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149 | if(l) |
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150 | { |
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151 | J=I; |
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152 | p=I[i]; |
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153 | J[i]=0; |
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154 | J=simplify(J,2); |
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155 | attrib(J,"isSB",1); |
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156 | pr=splitS(std(J,p),s,re); |
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157 | re=pr[1]; |
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158 | qr=insResult(qr,pr[2]); |
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159 | } |
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160 | else |
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161 | { |
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162 | J=interred(I); |
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163 | pr=splitS(timeStd(J,s),s,re); |
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164 | re=pr[1]; |
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165 | qr=insResult(qr,pr[2]); |
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166 | } |
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167 | } |
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168 | } |
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169 | return(list(re,qr)); |
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170 | } |
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171 | J=timeStd(I,s); //J=std(I) in splitS |
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172 | attrib(I,"isSB",1); |
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173 | if(size(reduce(J,I))==0){return(list(re+list(I),qr));} |
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174 | pr=splitS(J,s,re); |
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175 | return(list(re+pr[1],pr[2])); |
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176 | } |
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177 | example |
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178 | { "EXAMPLE:"; echo = 2; |
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179 | ring r=0,(b,s,t,u,v,w,x,y,z),dp; |
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180 | ideal i= |
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181 | bv+su, |
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182 | bw+tu, |
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183 | sw+tv, |
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184 | by+sx, |
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185 | bz+tx, |
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186 | sz+ty, |
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187 | uy+vx, |
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188 | uz+wx, |
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189 | vz+wy, |
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190 | bvz; |
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191 | splitS1(i,5); |
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192 | } |
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193 | |
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194 | /////////////////////////////////////////////////////////////////////////////// |
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195 | static proc splitS(ideal I,int s,list #) |
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196 | "USAGE: splitS(I,s[,l]) I ideal, s integer, l list of ideals |
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197 | COMPUTE: Factorizes the generators of I. If for one generator f=f1*f2 and |
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198 | fi=ni*gi^ri, ni an integer and gi a polynomial, then |
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199 | compute a standardbasis of I1=I,g1 and I2=I,g2, if this can be |
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200 | done in < s seconds. Then apply splitS to I1 and I2 and continue |
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201 | in the same way. The procedure stops if no generator can be |
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202 | factorized. |
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203 | RETURN: A list L of ideals and prime numbers |
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204 | L[1]: A list of ideals such that the radical of the intersection |
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205 | of these ideals coincides with the radical of I |
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206 | If the optional list l of the input is not empty then |
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207 | the ideals of L[1] which contain an ideal of l are canceled. |
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208 | L[2]: A list of prime numbers appearing as factors of the ni. |
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209 | NOTE: The computation avoids division by integers (by using |
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210 | option(contentSB)) hence the result is correct modulo any prime |
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211 | number which does not appear in the list L[2]. |
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212 | EXAMPLE: example splitS; shows an example |
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213 | " |
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214 | { |
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215 | option(redSB); |
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216 | option(contentSB); |
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217 | int j,k,e; |
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218 | int i=1; |
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219 | int l=attrib(I,"isSB"); |
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220 | ideal J; |
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221 | list re,fac,te,pr,qr,w; |
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222 | number n; |
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223 | poly p; |
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224 | re=#; |
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225 | |
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226 | if(deg(I[1])==0){return(list(re+list(std(I)),qr));} |
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227 | |
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228 | fac=factorize(I[1]); |
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229 | |
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230 | while((size(fac[1])==2)&&(i<size(I))) |
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231 | { |
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232 | i++; |
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233 | fac=factorize(I[i]); |
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234 | } |
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235 | if(size(fac[1])>2) |
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236 | { |
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237 | w=squarefreeP(number(fac[1][1])); |
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238 | n=w[1]; |
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239 | qr=insResult(qr,w[2]); |
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240 | for(j=2;j<=size(fac[1]);j++) |
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241 | { |
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242 | I[i]=fac[1][j]; |
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243 | attrib(I,"isSB",1); |
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244 | e=1; |
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245 | k=0; |
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246 | while(k<size(re)) |
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247 | { |
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248 | k++; |
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249 | if(size(reduce(re[k],I))==0){e=0;break;} |
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250 | attrib(re[k],"isSB",1); |
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251 | if(size(reduce(I,re[k]))==0){re=delete(re,k);k--;} |
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252 | } |
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253 | if(j==2){I[i]=I[i]*n;} |
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254 | if(e) |
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255 | { |
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256 | if(l) |
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257 | { |
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258 | J=I; |
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259 | p=I[i]; |
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260 | J[i]=0; |
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261 | J=simplify(J,2); |
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262 | attrib(J,"isSB",1); |
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263 | pr=splitS(std(J,p),s,re); |
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264 | re=pr[1]; |
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265 | qr=insResult(qr,pr[2]); |
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266 | } |
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267 | else |
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268 | { |
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269 | J=interred(I); |
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270 | pr=splitS(timeStd(J,s),s,re); |
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271 | re=pr[1]; |
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272 | qr=insResult(qr,pr[2]); |
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273 | } |
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274 | } |
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275 | } |
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276 | return(list(re,qr)); |
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277 | } |
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278 | J=std(I); |
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279 | attrib(I,"isSB",1); |
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280 | if(size(reduce(J,I))==0){return(list(re+list(I),qr));} |
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281 | pr=splitS(J,s,re); |
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282 | return(list(re+pr[1],pr[2])); |
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283 | } |
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284 | example |
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285 | { "EXAMPLE:"; echo = 2; |
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286 | ring r=0,(b,s,t,u,v,w,x,y,z),dp; |
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287 | ideal i= |
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288 | bv+su, |
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289 | bw+tu, |
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290 | sw+tv, |
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291 | by+sx, |
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292 | bz+tx, |
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293 | sz+ty, |
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294 | uy+vx, |
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295 | uz+wx, |
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296 | vz+wy, |
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297 | bvz; |
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298 | splitS(i,5); |
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299 | } |
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300 | |
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301 | /////////////////////////////////////////////////////////////////////////////// |
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302 | static proc finalSplit(list I,list qr) |
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303 | "USAGE: finalSplit(I,qr) I list of ideals, qr list of primes |
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304 | RETURN: list l of primes <=32003 such that 1 is in I[j]Z/p[x1..xn] |
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305 | for p not in l and all j or an ERROR if this is wrong or |
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306 | not decided. |
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307 | EXAMPLE: example finalSplit; shows an example |
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308 | " |
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309 | { |
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310 | option(redThrough); |
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311 | option(contentSB); |
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312 | ideal J,K; |
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313 | int i,j,k,n; |
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314 | int count=1; |
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315 | list q,pr,l; |
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316 | |
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317 | "trivial splitting"; |
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318 | pr=trivialSplit(I,4); |
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319 | l=pr[1]; |
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320 | qr=insResult(qr,pr[2]); |
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321 | while(count) |
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322 | { |
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323 | k++; |
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324 | "loop";k;size(l);"======"; |
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325 | count=0;list p; |
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326 | for(i=1;i<=size(l);i++) |
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327 | { |
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328 | i; |
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329 | K=changeOrdTest(l[i]); |
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330 | if(deg(K[1])!=0) |
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331 | { |
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332 | "split"; |
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333 | pr=splitS1(shortid(K,3),2); |
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334 | q=pr[1]; |
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335 | qr=insResult(qr,pr[2]); |
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336 | size(q);"out"; |
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337 | for(j=1;j<=size(q);j++) |
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338 | { |
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339 | if(deg(q[j][1])==0) |
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340 | { |
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341 | pr=contentS(q[j]); |
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342 | q[j]=pr[1]; |
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343 | qr=insResult(qr,pr[2]); |
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344 | } |
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345 | else |
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346 | { |
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347 | pr=simpliFy(q[j]+K); |
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348 | q[j]=pr[1]; |
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349 | qr=insResult(qr,pr[2]); |
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350 | pr=contentS(q[j]); |
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351 | q[j]=pr[1]; |
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352 | qr=insResult(qr,pr[2]); |
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353 | } |
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354 | } |
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355 | if((size(q)==1)&&(deg(q[1][1])>0)) |
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356 | { |
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357 | "split again"; |
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358 | K=sort(q[1])[1]; |
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359 | J=K[1..20+4*k]; |
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360 | pr=splitS(J,5); |
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361 | q=pr[1]; |
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362 | qr=insResult(qr,pr[2]); |
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363 | size(q);"out"; |
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364 | for(j=1;j<=size(q);j++) |
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365 | { |
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366 | if(deg(q[j][1])==0) |
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367 | { |
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368 | pr=contentS(q[j]); |
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369 | q[j]=pr[1]; |
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370 | qr=insResult(qr,pr[2]); |
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371 | } |
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372 | else |
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373 | { |
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374 | pr=simpliFy(q[j]+K); |
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375 | q[j]=pr[1]; |
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376 | qr=insResult(qr,pr[2]); |
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377 | pr=contentS(q[j]); |
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378 | q[j]=pr[1]; |
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379 | qr=insResult(qr,pr[2]); |
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380 | } |
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381 | } |
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382 | |
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383 | } |
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384 | if((size(q)==1)&&(deg(q[1][1])>0)) |
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385 | { |
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386 | n++; |
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387 | "split again"; |
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388 | K=q[1]; |
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389 | J=shortid(K,3+n); |
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390 | pr=splitS1(J,5); |
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391 | if(size(pr[1])>1) |
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392 | { |
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393 | q=pr[1]; |
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394 | qr=insResult(qr,pr[2]); |
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395 | size(q);"out"; |
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396 | for(j=1;j<=size(q);j++) |
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397 | { |
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398 | if(deg(q[j][1])==0) |
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399 | { |
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400 | pr=contentS(q[j]); |
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401 | q[j]=pr[1]; |
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402 | qr=insResult(qr,pr[2]); |
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403 | } |
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404 | else |
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405 | { |
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406 | pr=simpliFy(q[j]+K); |
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407 | q[j]=pr[1]; |
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408 | qr=insResult(qr,pr[2]); |
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409 | pr=contentS(q[j]); |
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410 | q[j]=pr[1]; |
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411 | qr=insResult(qr,pr[2]); |
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412 | } |
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413 | } |
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414 | } |
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415 | } |
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416 | if(size(q)>1) |
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417 | { |
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418 | count++; |
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419 | } |
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420 | else |
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421 | { |
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422 | if(deg(q[1][1])>0) |
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423 | { |
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424 | q[1]=changeOrdTest(q[1]); |
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425 | } |
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426 | } |
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427 | p=p+q; |
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428 | } |
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429 | else |
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430 | { |
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431 | pr=primefactors(number(K[1])); |
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432 | qr=insResult(qr,pr[1]); |
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433 | } |
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434 | } |
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435 | l=p; |
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436 | kill p; |
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437 | } |
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438 | l; |
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439 | for(i=1;i<=size(l);i++) |
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440 | { |
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441 | |
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442 | if(deg(l[i][1])>0){l;ERROR("not ready");} |
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443 | pr=primefactors(number(l[i][1])); |
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444 | if(pr[3]!=1){pr;ERROR("not ready");} |
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445 | qr=insResult(qr,pr[1]); |
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446 | } |
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447 | return(qr); |
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448 | } |
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449 | example |
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450 | { "EXAMPLE:"; echo = 2; |
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451 | ring r=0,x,dp; |
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452 | list qr=2,5,7; |
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453 | ideal i=181x-181,11x+11; |
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454 | list pr=i; |
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455 | finalSplit(pr,qr); |
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456 | } |
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457 | |
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458 | /////////////////////////////////////////////////////////////////////////////// |
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459 | proc noSolution(ideal I) |
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460 | "USAGE: noSolution(I) I ideal |
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461 | RETURN: list l of primes <=32003 such that 1 is in IZ/p[x1..xn] |
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462 | for p not in l or an ERROR if this is wrong or not decided. |
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463 | EXAMPLE: example noSolution; shows an example |
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464 | " |
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465 | { |
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466 | int s=1; |
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467 | int t=30; |
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468 | option(redThrough); |
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469 | option(contentSB); |
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470 | ideal J=shortid(I,3); |
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471 | ideal K; |
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472 | number n; |
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473 | |
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474 | "first splitting"; |
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475 | int i,j,k; |
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476 | list l,p,q,re,pr,qr; |
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477 | pr=splitS(J,s); |
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478 | l=pr[1]; |
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479 | qr=pr[2]; |
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480 | size(l); |
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481 | |
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482 | for(i=1;i<=size(l);i++) |
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483 | { |
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484 | if(deg(l[i][1])==0) |
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485 | { |
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486 | K=l[i][1]; |
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487 | } |
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488 | else |
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489 | { |
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490 | pr=simpliFy(I+l[i]); |
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491 | J=pr[1]; |
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492 | qr=insResult(qr,pr[2]); |
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493 | pr=contentS(J); |
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494 | J=pr[1]; |
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495 | qr=insResult(qr,pr[2]); |
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496 | K=timeStd(J,1); |
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497 | } |
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498 | if(deg(K[1])==0) |
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499 | { |
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500 | n=number(K[1]); |
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501 | pr=primefactors(n); |
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502 | if(pr[3]!=1) |
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503 | { |
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504 | J=changeOrdTest(J); |
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505 | p=p+list(J); |
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506 | } |
---|
507 | else |
---|
508 | { |
---|
509 | qr=insResult(qr,pr[1]); |
---|
510 | } |
---|
511 | } |
---|
512 | else |
---|
513 | { |
---|
514 | pr=contentS(K); |
---|
515 | p=p+list(pr[1]); |
---|
516 | qr=insResult(qr,pr[2]); |
---|
517 | } |
---|
518 | } |
---|
519 | "trivial splitting 1"; |
---|
520 | pr=trivialSplit(p,3); |
---|
521 | p=pr[1]; |
---|
522 | qr=insResult(qr,pr[2]); |
---|
523 | size(p); |
---|
524 | "second splitting"; |
---|
525 | |
---|
526 | for(i=1;i<=size(p);i++) |
---|
527 | { |
---|
528 | i; |
---|
529 | if(size(p[i])<t) |
---|
530 | { |
---|
531 | K=timeStd(p[i],10); |
---|
532 | if(deg(K[1])==0) |
---|
533 | { |
---|
534 | n=number(K[1]); |
---|
535 | pr=primefactors(n); |
---|
536 | if(pr[3]!=1) |
---|
537 | { |
---|
538 | p[i]=changeOrdTest(p[i]); |
---|
539 | re=re+list(p[i]); |
---|
540 | } |
---|
541 | else |
---|
542 | { |
---|
543 | qr=insResult(qr,pr[1]); |
---|
544 | } |
---|
545 | } |
---|
546 | else |
---|
547 | { |
---|
548 | pr=contentS(K); |
---|
549 | re=re+list(pr[1]); |
---|
550 | qr=insResult(qr,pr[2]); |
---|
551 | } |
---|
552 | } |
---|
553 | else |
---|
554 | { |
---|
555 | J=p[i]; |
---|
556 | J=J[1..t]; |
---|
557 | "in splitting"; |
---|
558 | pr=splitS(J,s); |
---|
559 | l=pr[1]; |
---|
560 | qr=insResult(qr,pr[2]); |
---|
561 | size(l); |
---|
562 | "out"; |
---|
563 | for(k=1;k<=size(l);k++) |
---|
564 | { |
---|
565 | pr=contentS(l[k]); |
---|
566 | l[k]= pr[1]; |
---|
567 | qr=insResult(qr,pr[2]); |
---|
568 | pr=simpliFy(l[k]+p[i]); |
---|
569 | J=pr[1]; |
---|
570 | qr=insResult(qr,pr[2]); |
---|
571 | K=timeStd(J,10); |
---|
572 | if(deg(K[1])==0) |
---|
573 | { |
---|
574 | n=number(K[1]); |
---|
575 | pr=primefactors(n); |
---|
576 | if(pr[3]!=1) |
---|
577 | { |
---|
578 | pr=contentS(J); |
---|
579 | qr=insResult(qr,pr[2]); |
---|
580 | J=pr[1]; |
---|
581 | J=changeOrdTest(J); |
---|
582 | re=re+list(J); |
---|
583 | } |
---|
584 | else |
---|
585 | { |
---|
586 | qr=insResult(qr,pr[1]); |
---|
587 | } |
---|
588 | } |
---|
589 | else |
---|
590 | { |
---|
591 | pr=contentS(K); |
---|
592 | qr=insResult(qr,pr[2]); |
---|
593 | re=re+list(pr[1]); |
---|
594 | } |
---|
595 | } |
---|
596 | } |
---|
597 | } |
---|
598 | "trivial splitting 2"; |
---|
599 | size(re); |
---|
600 | pr=trivialSplit(re,2); |
---|
601 | re=pr[1]; |
---|
602 | qr=insResult(qr,pr[2]); |
---|
603 | "third splitting"; |
---|
604 | size(re); |
---|
605 | for(i=1;i<=size(re);i++) |
---|
606 | { |
---|
607 | if(deg(re[i][1])>0) |
---|
608 | { |
---|
609 | i; |
---|
610 | pr=simpliFy(re[i]); |
---|
611 | J=pr[1]; |
---|
612 | qr=insResult(qr,pr[2]); |
---|
613 | "in splitting"; |
---|
614 | J=shortid(J,3); |
---|
615 | pr=splitS(J,s); |
---|
616 | l=pr[1]; |
---|
617 | if(size(l)==1) |
---|
618 | { |
---|
619 | "split again"; |
---|
620 | pr=simpliFy(re[i]); |
---|
621 | J=pr[1]; |
---|
622 | qr=insResult(qr,pr[2]); |
---|
623 | J=J[1..t]; |
---|
624 | pr=splitS(J,s); |
---|
625 | l=pr[1]; |
---|
626 | qr=insResult(qr,pr[2]); |
---|
627 | } |
---|
628 | else |
---|
629 | { |
---|
630 | qr=insResult(qr,pr[2]); |
---|
631 | } |
---|
632 | size(l); |
---|
633 | "out"; |
---|
634 | for(j=1;j<=size(l);j++) |
---|
635 | { |
---|
636 | pr=contentS(l[j]); |
---|
637 | l[j]=pr[1]; |
---|
638 | qr=insResult(qr,pr[2]); |
---|
639 | pr=simpliFy(re[i]+l[j]); |
---|
640 | J=pr[1]; |
---|
641 | qr=insResult(qr,pr[2]); |
---|
642 | K=timeStd(J,10); |
---|
643 | if(deg(K[1])==0) |
---|
644 | { |
---|
645 | n=number(K[1]); |
---|
646 | pr=primefactors(n); |
---|
647 | if(pr[3]!=1) |
---|
648 | { |
---|
649 | pr=contentS(J); |
---|
650 | J=pr[1]; |
---|
651 | qr=insResult(qr,pr[2]); |
---|
652 | J=changeOrdTest(J); |
---|
653 | q=q+list(J); |
---|
654 | } |
---|
655 | else |
---|
656 | { |
---|
657 | qr=insResult(qr,pr[1]); |
---|
658 | } |
---|
659 | } |
---|
660 | else |
---|
661 | { |
---|
662 | pr=contentS(K); |
---|
663 | qr=insResult(qr,pr[2]); |
---|
664 | q=q+list(pr[1]); |
---|
665 | } |
---|
666 | } |
---|
667 | } |
---|
668 | else |
---|
669 | { |
---|
670 | i; |
---|
671 | pr=primefactors(number(re[i][1])); |
---|
672 | qr=insResult(qr,pr[1]); |
---|
673 | } |
---|
674 | } |
---|
675 | "jetzt geht es los"; |
---|
676 | return(finalSplit(q,qr)); |
---|
677 | } |
---|
678 | example |
---|
679 | { "EXAMPLE:"; echo = 2; |
---|
680 | ring r=0,x,dp; |
---|
681 | ideal i=181x-181,11x+11; |
---|
682 | noSolution(i); |
---|
683 | } |
---|
684 | |
---|
685 | /////////////////////////////////////////////////////////////////////////////// |
---|
686 | |
---|
687 | static proc changeOrdTest(ideal I) |
---|
688 | { |
---|
689 | def R=basering; |
---|
690 | if(deg(I[1])==0){return(I);} |
---|
691 | def RR=changeord("dp"); |
---|
692 | setring RR; |
---|
693 | ideal I=imap(R,I); |
---|
694 | ideal K=timeStd(I,5); |
---|
695 | if(deg(K[1])==0) |
---|
696 | { |
---|
697 | number n=number(K[1]); |
---|
698 | if(primefactors(n)[3]==1) |
---|
699 | { |
---|
700 | I=K; |
---|
701 | } |
---|
702 | } |
---|
703 | setring R; |
---|
704 | I=imap(RR,I); |
---|
705 | kill RR; |
---|
706 | return(I); |
---|
707 | } |
---|
708 | /////////////////////////////////////////////////////////////////////////////// |
---|
709 | |
---|
710 | static proc trivialSplit(list p,int depth, list #) |
---|
711 | "USAGE: trivialSplit(p,s) p list of ideals, s integer the number of iterations |
---|
712 | COMPUTE: Factorizes the monomials among the generators of I. |
---|
713 | If one monomial contains the variables x1,..xr, then |
---|
714 | I1=I(x1=0),...,Ir=I(xr=0) is considered. Then apply |
---|
715 | trivialSplit to I1 ...Ir and continue in the same way s times. |
---|
716 | RETURN: A list L of ideals and prime numbers |
---|
717 | L[1]: A list of ideals such that the radical of the intersection |
---|
718 | of these ideals at x1=...=xr=0 coincides with the radical of |
---|
719 | I at x1=...=xr=0. |
---|
720 | L[2]: A list of prime numbers appearing as factors of the monomials. |
---|
721 | NOTE: The computation avoids division by integers hence the result |
---|
722 | is correct modulo any prime |
---|
723 | number which does not appear in the list L[2]. |
---|
724 | EXAMPLE: example trivialSplit; shows an example |
---|
725 | " |
---|
726 | { |
---|
727 | list re,l,pr,qr; |
---|
728 | int i,k; |
---|
729 | ideal J,K,T,Ke; |
---|
730 | number n; |
---|
731 | |
---|
732 | if(size(p)==0){return(list(p,qr));} |
---|
733 | if(depth<=0){return(list(p,qr));} |
---|
734 | if(size(#)>0) |
---|
735 | { |
---|
736 | T=#[1]; |
---|
737 | } |
---|
738 | else |
---|
739 | { |
---|
740 | T=1; |
---|
741 | } |
---|
742 | for(k=1;k<=size(p);k++) |
---|
743 | { |
---|
744 | pr=simpliFy(p[k]); |
---|
745 | p[k]=pr[1]; |
---|
746 | qr=insResult(qr,pr[2]); |
---|
747 | J=shortid(p[k],1); |
---|
748 | if((size(J)>0)&&(deg(J[1])>=1)) |
---|
749 | { |
---|
750 | pr=splitS1(J,10); |
---|
751 | l=pr[1]; |
---|
752 | qr=insResult(qr,pr[2]); |
---|
753 | for(i=1;i<=size(l);i++) |
---|
754 | { |
---|
755 | Ke=l[i]; |
---|
756 | l[i]=trivialSimplify(p[k],l[i]); |
---|
757 | pr=simpliFy(l[i]); |
---|
758 | l[i]=pr[1]; |
---|
759 | qr=insResult(qr,pr[2]); |
---|
760 | K=timeStd(l[i],1); |
---|
761 | attrib(K,"isSB",1); |
---|
762 | if(deg(K[1])==0) |
---|
763 | { |
---|
764 | n=number(K[1]); |
---|
765 | if(primefactors(n)[3]!=1) |
---|
766 | { |
---|
767 | l[i]=changeOrdTest(l[i]); |
---|
768 | pr=trivialSplit(l[i],depth-1,T); |
---|
769 | re=re+pr[1]; |
---|
770 | qr=insResult(qr,pr[2]); |
---|
771 | } |
---|
772 | else |
---|
773 | { |
---|
774 | l[i]=K; //neu |
---|
775 | } |
---|
776 | } |
---|
777 | else |
---|
778 | { |
---|
779 | if(size(reduce(trivialSimplify(T,Ke),K))!=0) |
---|
780 | { |
---|
781 | pr=trivialSplit(K,depth-1,trivialSimplify(T,Ke)); |
---|
782 | re=re+pr[1]; |
---|
783 | qr=insResult(qr,pr[2]); |
---|
784 | } |
---|
785 | } |
---|
786 | T=intersect(T,Ke); |
---|
787 | } |
---|
788 | } |
---|
789 | else |
---|
790 | { |
---|
791 | J=timeStd(p[k],5); |
---|
792 | if(deg(J[1])==0) |
---|
793 | { |
---|
794 | n=number(J[1]); |
---|
795 | if(primefactors(n)[3]!=1) |
---|
796 | { |
---|
797 | p[k]=changeOrdTest(p[k]); |
---|
798 | re=re+list(p[k]); |
---|
799 | } |
---|
800 | else |
---|
801 | { |
---|
802 | re=re+list(J); |
---|
803 | } |
---|
804 | } |
---|
805 | else |
---|
806 | { |
---|
807 | re=re+list(J); |
---|
808 | } |
---|
809 | } |
---|
810 | } |
---|
811 | return(list(re,qr)); |
---|
812 | } |
---|
813 | example |
---|
814 | { "EXAMPLE:"; echo = 2; |
---|
815 | ring r=0,(b,s,t,u,v,w,x,y,z),dp; |
---|
816 | ideal i= |
---|
817 | bv+su, |
---|
818 | bw+tu, |
---|
819 | sw+tv, |
---|
820 | by+sx, |
---|
821 | bz+tx, |
---|
822 | sz+ty, |
---|
823 | uy+vx, |
---|
824 | uz+wx, |
---|
825 | vz+wy, |
---|
826 | bvz; |
---|
827 | trivialSplit(i,2); |
---|
828 | } |
---|
829 | |
---|
830 | /////////////////////////////////////////////////////////////////////////////// |
---|
831 | |
---|
832 | static proc trivialSimplify(ideal I, ideal J) |
---|
833 | "USAGE: trivialSimplify(I,J); I,J ideals, J generated by variables |
---|
834 | RETURN: ideal K = eliminate(I,m) m the product of variables of J |
---|
835 | EXAMPLE: example trivialSimplify; shows an example |
---|
836 | " |
---|
837 | { |
---|
838 | int i; |
---|
839 | for(i=1;i<=size(J);i++){I=subst(I,J[i],0);} |
---|
840 | return(simplify(I,2)); |
---|
841 | } |
---|
842 | example |
---|
843 | { "EXAMPLE:"; echo = 2; |
---|
844 | ring r=0,(x,y,z,w,t,u),dp; |
---|
845 | ideal i= |
---|
846 | t,u, |
---|
847 | x3+y2+2z, |
---|
848 | x2+3y, |
---|
849 | x2+y2+z2, |
---|
850 | w2+z+u; |
---|
851 | ideal j=t,u; |
---|
852 | trivialSimplify(i,j); |
---|
853 | } |
---|
854 | |
---|
855 | /////////////////////////////////////////////////////////////////////////////// |
---|
856 | |
---|
857 | static proc simpliFy(ideal J,list #) |
---|
858 | "USAGE: simpliFy(id); id ideal |
---|
859 | RETURN: ideal I = eliminate(Id,m) m is a product of variables |
---|
860 | which are in linear equations involved in the generators of id |
---|
861 | EXAMPLE: example simpliFy; shows an example |
---|
862 | " |
---|
863 | { |
---|
864 | ideal I=J; |
---|
865 | if(size(#)!=0){I=#[1];} |
---|
866 | def R=basering; |
---|
867 | matrix M=jacob(I); |
---|
868 | ideal ma=maxideal(1); |
---|
869 | int i,j,k; |
---|
870 | map phi; |
---|
871 | list pr,re; |
---|
872 | |
---|
873 | for(i=1;i<=nrows(M);i++) |
---|
874 | { |
---|
875 | for(j=1;j<=ncols(M);j++) |
---|
876 | { |
---|
877 | if((deg(M[i,j])==0)&&(deg(I[i])==1)) |
---|
878 | { |
---|
879 | ma[j]=(-1/M[i,j])*(I[i]-M[i,j]*var(j)); |
---|
880 | pr=primefactors(number(M[i,j])); |
---|
881 | if(pr[3]>1){pr[3];ERROR("number too big in simpliFy");} |
---|
882 | re=insResult(re,pr[1]); |
---|
883 | phi=R,ma; |
---|
884 | I=phi(I); |
---|
885 | J=phi(J); |
---|
886 | for(k=1;k<=ncols(I);k++) |
---|
887 | { |
---|
888 | I[k]=cleardenom(I[k]); |
---|
889 | } |
---|
890 | M=jacob(I); |
---|
891 | } |
---|
892 | } |
---|
893 | } |
---|
894 | J=simplify(J,2); |
---|
895 | for(i=1;i<=size(J);i++) |
---|
896 | { |
---|
897 | if(deg(J[i])==0){J=std(J);break;} |
---|
898 | J[i]=cleardenom(J[i]); |
---|
899 | } |
---|
900 | return(list(J,re)); |
---|
901 | } |
---|
902 | example |
---|
903 | { "EXAMPLE:"; echo = 2; |
---|
904 | ring r=0,(x,y,z,w),dp; |
---|
905 | ideal i= |
---|
906 | x3+y2+2z, |
---|
907 | x2+3y, |
---|
908 | x2+y2+z2, |
---|
909 | w2+z; |
---|
910 | simpliFy(i); |
---|
911 | } |
---|
912 | /////////////////////////////////////////////////////////////////////////////// |
---|
913 | |
---|
914 | static proc squarefreeP(number n) |
---|
915 | "USAGE: squarefreeP(n); n number |
---|
916 | RETURN: list l of numbers. l[2] contains all prime factors of n less |
---|
917 | then 32003, l[1] is the part of m which has prime factors greater |
---|
918 | then 32003 |
---|
919 | EXAMPLE: example squarefreeP; shows an example |
---|
920 | " |
---|
921 | { |
---|
922 | list re; |
---|
923 | if(n<0){n=-n;} |
---|
924 | if(n==1){return(list(n,re));} |
---|
925 | list pr=primefactors(n); |
---|
926 | int i; |
---|
927 | number m=number(pr[3][1]); |
---|
928 | re=insResult(re,pr[1]); |
---|
929 | return(list(m,re)); |
---|
930 | } |
---|
931 | example |
---|
932 | { "EXAMPLE:"; echo = 2; |
---|
933 | ring r = 0,x,dp; |
---|
934 | squarefreeP(123456789100); |
---|
935 | } |
---|
936 | |
---|
937 | /////////////////////////////////////////////////////////////////////////////// |
---|
938 | |
---|
939 | static proc contentS(ideal I) |
---|
940 | "USAGE: contentS(I); I ideal |
---|
941 | RETURN: list l |
---|
942 | l[1] the ideal I. Te generators of I are the generators of the |
---|
943 | input ideal devided by the part of the content which have |
---|
944 | prime factors less then 32003. |
---|
945 | l[2] contains the prime numbers which occured in the division |
---|
946 | EXAMPLE: example contentS; shows an example |
---|
947 | " |
---|
948 | |
---|
949 | { |
---|
950 | option(contentSB); |
---|
951 | int i,k; |
---|
952 | number n,m; |
---|
953 | list pr,re; |
---|
954 | for(i=1;i<=size(I);i++) |
---|
955 | { |
---|
956 | n=leadcoef(I[i]); |
---|
957 | if(n<0){n=-n;} |
---|
958 | if(n>1) |
---|
959 | { |
---|
960 | pr=primefactors(n); |
---|
961 | if(n<=32003) |
---|
962 | { |
---|
963 | m=n; |
---|
964 | } |
---|
965 | else |
---|
966 | { |
---|
967 | m=number(pr[1][1])^pr[2][1]; |
---|
968 | for(k=2;k<=size(pr[1]);k++) |
---|
969 | { |
---|
970 | m=m*number(pr[1][k])^pr[2][k]; |
---|
971 | } |
---|
972 | } |
---|
973 | I[i]=cleardenom(I[i]/m); |
---|
974 | re=insResult(re,pr[1]); |
---|
975 | } |
---|
976 | } |
---|
977 | return(list(I,re)); |
---|
978 | } |
---|
979 | example |
---|
980 | { "EXAMPLE:"; echo = 2; |
---|
981 | ring r = 0,(x,y),dp; |
---|
982 | ideal I=2x+2,3y+3x,1234567891x+1234567891; |
---|
983 | contentS(I); |
---|
984 | } |
---|
985 | |
---|
986 | /////////////////////////////////////////////////////////////////////////////// |
---|
987 | |
---|
988 | static proc insResult(list r,#) |
---|
989 | { |
---|
990 | if(size(#)==0){return(r);} |
---|
991 | if(size(r)==0){r[1]=1;} |
---|
992 | int i,j; |
---|
993 | for(i=1;i<=size(#);i++) |
---|
994 | { |
---|
995 | j=1; |
---|
996 | while((#[i]>r[j])&&(j<size(r))){j++;} |
---|
997 | if(#[i]>r[j]){r=insert(r,#[i],j);} |
---|
998 | if(#[i]<r[j]){r=insert(r,#[i],j-1);} |
---|
999 | } |
---|
1000 | return(r); |
---|
1001 | } |
---|
1002 | example |
---|
1003 | { "EXAMPLE:"; echo = 2; |
---|
1004 | list r=list(3,7,13); |
---|
1005 | intvec v=2,3,5,11,17; |
---|
1006 | insResult(r,v); |
---|
1007 | } |
---|
1008 | /////////////////////////////////////////////////////////////////////////////// |
---|
1009 | |
---|
1010 | static proc shortid (ideal id,int n) |
---|
1011 | { |
---|
1012 | ideal Lc; |
---|
1013 | intvec v; |
---|
1014 | int ii; |
---|
1015 | for(ii= 1; ii<=ncols(id); ii++) |
---|
1016 | { |
---|
1017 | if (size(id[ii])<=n and id[ii]!=0) |
---|
1018 | { |
---|
1019 | Lc = Lc,id[ii]; |
---|
1020 | v=v,ii; |
---|
1021 | } |
---|
1022 | } |
---|
1023 | Lc = simplify(Lc,2); |
---|
1024 | list L = Lc,v; |
---|
1025 | return(Lc); |
---|
1026 | } |
---|
1027 | |
---|
1028 | |
---|
1029 | /* |
---|
1030 | //-------------- the solution of the problem ----------------- |
---|
1031 | ring r = 0,(b,c,t),dp; //global (affine) ring |
---|
1032 | matrix X[2][2] = 0, -1, |
---|
1033 | 1, t; |
---|
1034 | matrix Y[2][2] = 1, b, |
---|
1035 | c, 1+bc; |
---|
1036 | ; |
---|
1037 | |
---|
1038 | //------------ create word W1 and ideal I of U1-U2 ------------ |
---|
1039 | matrix iX = inverse(X); |
---|
1040 | matrix iY = inverse(Y); |
---|
1041 | matrix U1 = iX*Y*X*iY*X; //the word w1 |
---|
1042 | //matrix U1=iX*iX*iY*X; //das neue Wort |
---|
1043 | matrix N = X*U1*iX; |
---|
1044 | matrix M = Y*U1*iY; |
---|
1045 | matrix iN = inverse(N); |
---|
1046 | matrix iM = inverse(M); |
---|
1047 | matrix U2 = N*M*iN*iM; |
---|
1048 | ideal I = ideal(U1-U2); |
---|
1049 | |
---|
1050 | //list qr=primdecGTZ(I); |
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1051 | //I=qr[1][2]; |
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1052 | |
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1053 | ring rh = 0,(b,c,t,h),dp; |
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1054 | ideal I = imap(r,I); |
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1055 | ideal sI = groebner(I); |
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1056 | ideal hI = homog(sI,h); |
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1057 | ideal shI =std(hI); |
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1058 | ideal J = eliminate(shI,c); //eliminate c |
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1059 | poly f = J[1]; |
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1060 | |
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1061 | ring r1 = 0,(b,t,h),dp; |
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1062 | poly hf=b6t6-2b5t7+b4t8+b8t3h-4b7t4h+7b6t5h-6b5t6h+2b4t7h-7b6t4h2+12b5t5h2+b4t6h2-6b3t7h2-3b8th3+12b7t2h3-16b6t3h3+19b4t5h3-12b3t6h3-2b8h4+8b7th4-3b6t2h4+2b5t3h4-45b4t4h4+32b3t5h4+12b2t6h4-12b6th5+50b5t2h5-64b4t3h5+4b3t4h5+26b2t5h5-12b6h6+24b5th6+22b4t2h6+12b3t3h6-73b2t4h6-10bt5h6-8b5h7-6b4th7+88b3t2h7-68b2t3h7-26bt4h7-29b4h8+16b3th8+42b2t2h8+54bt3h8+3t4h8-28b3h9-12b2th9+88bt2h9+10t3h9-38b2h10-8bth10-11t2h10-28bh11-34th11-17h12; |
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1063 | //poly hf=b3t4-b2t5+b4t2h-2b3t3h+2b2t4h-bt5h-2b3t2h2+4bt4h2+b2t2h3-bt3h3+t4h3 |
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1064 | //+2b2th4-6bt2h4-4t3h4+b2h5-2bth5+2t2h5+4th6+h7; |
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1065 | |
---|
1066 | int n,m,sA,sB; |
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1067 | n=6; |
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1068 | //n=3; |
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1069 | //m=7-n; |
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1070 | |
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1071 | m = 12-n; |
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1072 | ideal A = maxideal(n); ideal B = maxideal(m); |
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1073 | sA =size(A); |
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1074 | sB = size(B); |
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1075 | |
---|
1076 | ring R = 0,(b(1..sB),a(1..sA),b,t,h),dp; |
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1077 | poly hf = imap(r1,hf); |
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1078 | ideal A = imap(r1,A); |
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1079 | ideal B = imap(r1,B); |
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1080 | matrix aa[sA][1]= a(1..sA); |
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1081 | matrix bb[sB][1] = b(1..sB); |
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1082 | poly f1= (matrix(A)*aa)[1,1]; //Ansatz for degree n |
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1083 | poly f2= (matrix(B)*bb)[1,1]; //Ansatz for degree m=12-n |
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1084 | poly g = f1*f2-hf; //assume hf factors |
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1085 | matrix C = coef(g,bth); |
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1086 | ideal co = submat(C,2,1..ncols(C));//condition for decomposition, size 91 |
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1087 | |
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1088 | ring R1 = 0,(b(1..sB),a(1..sA)),lp; |
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1089 | ideal co = imap(R,co); |
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1090 | co=subst(co,a(1),1); //OE a1=1 |
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1091 | co = subst(co,b(1),-17); //co1[91]=b(1)+17 |
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1092 | //co = subst(co,b(1),1); |
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1093 | |
---|
1094 | int aa=timer; |
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1095 | list pr=noSolution(co); |
---|
1096 | timer-aa; |
---|
1097 | |
---|
1098 | |
---|
1099 | //n=1 0 sec |
---|
1100 | // keine Primzahl |
---|
1101 | //n=2 628 sec |
---|
1102 | // 2,3,5,7,11,13,17,19,23,29,31,37,43,47,61,89,293,347,367,487,491,3463,7498 |
---|
1103 | //n=3 1604 sec |
---|
1104 | // 2,3,5,7,11,13,17,19,23,29,31,59,71,79,101,163,197,211,269,281,541,647,863, |
---|
1105 | // 7129,9041,10343,18413,20857 |
---|
1106 | //n=4 8296 sec |
---|
1107 | // 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,61,67,71,73,83,89,101,109,127, |
---|
1108 | 131,149,151,157,163,167,173,179,193,199,211,223,229,251,263,283,307,313,331, |
---|
1109 | 347,373,401,601,607,631,643,701,719,727,829,857,887,971,1279,1361,1453,1721, |
---|
1110 | 1847,2003,2069,2213,2671,2693,3299,3373,3391,3517,3593,3701,3779,4111,4409, |
---|
1111 | 4423,4561,4657,4793,5273,5399,5659,5987,6949,7487,8011,8243,8887,9769,10159, |
---|
1112 | 10177,12007,26347 |
---|
1113 | //n=5 4715 sec |
---|
1114 | // 2,3,5,7,11,13,17,19,23,31,41,43,61,283,421,631,1609 |
---|
1115 | //n=6 18317 sec |
---|
1116 | // 2,3,5,7,11,13,17,19,29,31,37,41,47,61,71,73,79,89,97,127,131,173,181,223, |
---|
1117 | // 269,953,1039,6151,6343,7823 |
---|
1118 | |
---|
1119 | |
---|
1120 | //fuer das andere Wort |
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1121 | //n=1 keine Primzahl |
---|
1122 | //n=2 2,3,7,109 |
---|
1123 | //n=3 2,3,5,7,11,13,19,29,31,41,43,47,89,139,149,167,173,991,1381,1637,27367 |
---|
1124 | |
---|
1125 | */ |
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