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14 <a name="package-MathUtil-"></a> 1 <span class="k">package </span><span class="i">MathUtil</span><span class="sc">;</span>
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15 2 <span class="c">#</span>
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16 3 <span class="c"># $RCSfile: MathUtil.pm,v $</span>
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17 4 <span class="c"># $Date: 2015/02/28 20:47:17 $</span>
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18 5 <span class="c"># $Revision: 1.28 $</span>
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19 6 <span class="c">#</span>
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20 7 <span class="c"># Author: Manish Sud <msud@san.rr.com></span>
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21 8 <span class="c">#</span>
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22 9 <span class="c"># Copyright (C) 2015 Manish Sud. All rights reserved.</span>
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23 10 <span class="c">#</span>
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24 11 <span class="c"># This file is part of MayaChemTools.</span>
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25 12 <span class="c">#</span>
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26 13 <span class="c"># MayaChemTools is free software; you can redistribute it and/or modify it under</span>
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27 14 <span class="c"># the terms of the GNU Lesser General Public License as published by the Free</span>
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28 15 <span class="c"># Software Foundation; either version 3 of the License, or (at your option) any</span>
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29 16 <span class="c"># later version.</span>
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30 17 <span class="c">#</span>
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31 18 <span class="c"># MayaChemTools is distributed in the hope that it will be useful, but without</span>
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32 19 <span class="c"># any warranty; without even the implied warranty of merchantability of fitness</span>
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33 20 <span class="c"># for a particular purpose. See the GNU Lesser General Public License for more</span>
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34 21 <span class="c"># details.</span>
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35 22 <span class="c">#</span>
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36 23 <span class="c"># You should have received a copy of the GNU Lesser General Public License</span>
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37 24 <span class="c"># along with MayaChemTools; if not, see <http://www.gnu.org/licenses/> or</span>
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38 25 <span class="c"># write to the Free Software Foundation Inc., 59 Temple Place, Suite 330,</span>
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39 26 <span class="c"># Boston, MA, 02111-1307, USA.</span>
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40 27 <span class="c">#</span>
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41 28
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42 29 <span class="k">use</span> <span class="w">strict</span><span class="sc">;</span>
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43 30 <span class="k">use</span> <span class="w">Exporter</span><span class="sc">;</span>
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44 31 <span class="k">use</span> <span class="w">Constants</span><span class="sc">;</span>
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45 32 <span class="k">use</span> <span class="w">Math::Trig</span> <span class="s">(</span><span class="s">)</span><span class="sc">;</span>
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46 33 <span class="k">use</span> <span class="w">POSIX</span> <span class="s">(</span><span class="s">)</span><span class="sc">;</span>
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47 34
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48 35 <span class="k">use</span> <span class="w">vars</span> <span class="q">qw(@ISA @EXPORT @EXPORT_OK %EXPORT_TAGS)</span><span class="sc">;</span>
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49 36
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50 37 <span class="i">@ISA</span> = <span class="q">qw(Exporter)</span><span class="sc">;</span>
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51 38 <span class="i">@EXPORT</span> = <span class="q">qw(acos asin atan tan ceil floor log10 min max srandom random round GeneratePrimeNumbersUpToLimit GeneratePrimeNumbersUpToCount)</span><span class="sc">;</span>
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52 39 <span class="i">@EXPORT_OK</span> = <span class="q">qw()</span><span class="sc">;</span>
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53 40
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54 41 <span class="i">%EXPORT_TAGS</span> = <span class="s">(</span><span class="w">all</span> <span class="cm">=></span> <span class="s">[</span><span class="i">@EXPORT</span><span class="cm">,</span> <span class="i">@EXPORT_OK</span><span class="s">]</span>
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55 42 <span class="s">)</span><span class="sc">;</span>
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56 43
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57 44
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58 45 <span class="c"># Return next largest integer...</span>
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59 <a name="ceil-"></a> 46 <span class="k">sub </span><span class="m">ceil ($)</span> <span class="s">{</span>
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60 47 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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61 48
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62 49 <span class="k">return</span> <span class="i">POSIX::ceil</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span><span class="sc">;</span>
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63 50 <span class="s">}</span>
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64 51
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65 52 <span class="c"># Return previous smallest integer...</span>
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66 <a name="floor-"></a> 53 <span class="k">sub </span><span class="m">floor ($)</span> <span class="s">{</span>
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67 54 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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68 55
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69 56 <span class="k">return</span> <span class="i">POSIX::floor</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span><span class="sc">;</span>
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70 57 <span class="s">}</span>
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71 58
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72 59 <span class="c"># Calculate log value using base 10...</span>
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73 <a name="log10-"></a> 60 <span class="k">sub </span><span class="m">log10 ($)</span> <span class="s">{</span>
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74 61 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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75 62
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76 63 <span class="k">return</span> <span class="i">CORE::</span><span class="k">log</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span>/<span class="i">CORE::</span><span class="k">log</span><span class="s">(</span><span class="n">10</span><span class="s">)</span><span class="sc">;</span>
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77 64 <span class="s">}</span>
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78 65
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79 66 <span class="c"># Return the smaller of two numbers...</span>
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80 <a name="min-"></a> 67 <span class="k">sub </span><span class="m">min ($$)</span> <span class="s">{</span>
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81 68 <span class="k">my</span><span class="s">(</span><span class="i">$Value1</span><span class="cm">,</span> <span class="i">$Value2</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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82 69
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83 70 <span class="k">return</span> <span class="s">(</span><span class="i">$Value1</span> <= <span class="i">$Value2</span><span class="s">)</span> ? <span class="i">$Value1</span> <span class="co">:</span> <span class="i">$Value2</span><span class="sc">;</span>
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84 71 <span class="s">}</span>
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85 72
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86 73 <span class="c"># Return the larger of two numbers...</span>
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87 <a name="max-"></a> 74 <span class="k">sub </span><span class="m">max ($$)</span> <span class="s">{</span>
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88 75 <span class="k">my</span><span class="s">(</span><span class="i">$Value1</span><span class="cm">,</span> <span class="i">$Value2</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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89 76
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90 77 <span class="k">return</span> <span class="s">(</span><span class="i">$Value1</span> >= <span class="i">$Value2</span><span class="s">)</span> ? <span class="i">$Value1</span> <span class="co">:</span> <span class="i">$Value2</span><span class="sc">;</span>
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91 78 <span class="s">}</span>
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92 79
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93 80 <span class="c"># The random number generator implemented in MayaChemTools is a variant of linear</span>
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94 81 <span class="c"># congruential generator (LCG) as described by Miller et al. [ Ref 120 ]. It is</span>
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95 82 <span class="c"># also referred to as Lehmer random number generator or Park-Miller random number</span>
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96 83 <span class="c"># generator.</span>
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97 84 <span class="c">#</span>
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98 85 <span class="c"># Unlike Perl's core random number generator function rand, the random number</span>
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99 86 <span class="c"># generator implemented in MayaChemTools generates consistent random values</span>
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100 87 <span class="c"># across different platforms - Windows, CygWin, Linux, Unix - for a specific random</span>
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101 88 <span class="c"># seed.</span>
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102 89 <span class="c">#</span>
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103 90
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104 91 <span class="c"># $RandomModulus = 2**31 - 1;</span>
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105 92 <span class="c"># $RandomMultiplier = 16807;</span>
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106 93 <span class="c"># $RandomQuotient = $RandomModulus / $RandomMultiplier;</span>
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107 94 <span class="c"># $RandomRemainder = $RandomModulus % $RandomMultiplier</span>
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108 95 <span class="c">#</span>
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109 96 <span class="c"># $MaxRandomSeed = 2*31 -2</span>
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110 97 <span class="c">#</span>
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111 98 <span class="k">my</span><span class="s">(</span><span class="i">$MaxRandomSeed</span><span class="cm">,</span> <span class="i">$RandomSeed</span><span class="cm">,</span> <span class="i">$RandomModulus</span><span class="cm">,</span> <span class="i">$RandomMultiplier</span><span class="cm">,</span> <span class="i">$RandomQuotient</span><span class="cm">,</span> <span class="i">$RandomRemainder</span><span class="s">)</span><span class="sc">;</span>
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112 99
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113 100 <span class="i">$MaxRandomSeed</span> = <span class="n">2147483646</span><span class="sc">;</span>
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114 101 <span class="i">$RandomSeed</span> = <span class="n">123456789</span><span class="sc">;</span>
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115 102
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116 103 <span class="i">$RandomModulus</span> = <span class="n">2147483647</span><span class="sc">;</span>
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117 104 <span class="i">$RandomMultiplier</span> = <span class="n">16807</span><span class="sc">;</span>
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118 105 <span class="i">$RandomQuotient</span> = <span class="n">127773</span><span class="sc">;</span>
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119 106 <span class="i">$RandomRemainder</span> = <span class="n">2836</span><span class="sc">;</span>
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120 107
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121 108 <span class="c"># Set random number seed...</span>
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122 109 <span class="c">#</span>
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123 110 <span class="c"># The intial value of random number seed is recommeded to be an integer between 1</span>
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124 111 <span class="c"># and 2**31 - 2 [Ref 120] which translates to be 1 and 2147483646</span>
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125 112 <span class="c">#</span>
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126 <a name="srandom-"></a> 113 <span class="k">sub </span><span class="m">srandom ($)</span> <span class="s">{</span>
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127 114 <span class="k">my</span><span class="s">(</span><span class="i">$Seed</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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128 115
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129 116 <span class="k">if</span> <span class="s">(</span><span class="i">$Seed</span> <= <span class="n">0</span> <span class="s">)</span> <span class="s">{</span>
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130 117 <span class="k">die</span> <span class="q">"Error: srandom: Specified seed value must be greater than 0..."</span><span class="sc">;</span>
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131 118 <span class="s">}</span>
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132 119
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133 120 <span class="i">$RandomSeed</span> = <span class="s">(</span><span class="i">$Seed</span> > <span class="i">$MaxRandomSeed</span><span class="s">)</span> ? <span class="s">(</span><span class="i">$Seed</span> % <span class="i">$MaxRandomSeed</span><span class="s">)</span> <span class="co">:</span> <span class="i">$Seed</span><span class="sc">;</span>
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134 121
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135 122 <span class="k">return</span> <span class="i">$RandomSeed</span><span class="sc">;</span>
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136 123 <span class="s">}</span>
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137 124
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138 125 <span class="c"># Retrun a random number between 0 and less than 1 or specified size...</span>
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139 126 <span class="c">#</span>
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140 <a name="random-"></a> 127 <span class="k">sub </span><span class="m">random (;$)</span> <span class="s">{</span>
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141 128 <span class="k">my</span><span class="s">(</span><span class="i">$Size</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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142 129 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="cm">,</span> <span class="i">$LowValue</span><span class="cm">,</span> <span class="i">$HighValue</span><span class="s">)</span><span class="sc">;</span>
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143 130
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144 131 <span class="i">$Size</span> = <span class="k">defined</span> <span class="i">$Size</span> ? <span class="i">$Size</span> <span class="co">:</span> <span class="n">1.0</span><span class="sc">;</span>
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145 132
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146 133 <span class="i">$HighValue</span> = <span class="i">$RandomSeed</span> / <span class="i">$RandomQuotient</span><span class="sc">;</span>
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147 134 <span class="i">$LowValue</span> = <span class="i">$RandomSeed</span> % <span class="i">$RandomQuotient</span><span class="sc">;</span>
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148 135
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149 136 <span class="i">$Value</span> = <span class="i">$RandomMultiplier</span> * <span class="i">$LowValue</span> - <span class="i">$RandomRemainder</span> * <span class="i">$HighValue</span><span class="sc">;</span>
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150 137
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151 138 <span class="i">$RandomSeed</span> = <span class="s">(</span><span class="i">$Value</span> > <span class="n">0</span><span class="s">)</span> ? <span class="i">$Value</span> <span class="co">:</span> <span class="s">(</span><span class="i">$Value</span> + <span class="i">$RandomModulus</span><span class="s">)</span><span class="sc">;</span>
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152 139
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153 140 <span class="k">return</span> <span class="s">(</span><span class="i">$RandomSeed</span> / <span class="i">$RandomModulus</span><span class="s">)</span> * <span class="i">$Size</span><span class="sc">;</span>
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154 141 <span class="s">}</span>
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155 142
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156 143 <span class="c"># Round a integer/real number to:</span>
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157 144 <span class="c"># . A nearest integer</span>
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158 145 <span class="c"># . Specified number of decimal places</span>
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159 146 <span class="c">#</span>
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160 <a name="round-"></a> 147 <span class="k">sub </span><span class="m">round ($;$)</span> <span class="s">{</span>
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161 148 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="cm">,</span> <span class="i">$DecimalPlaces</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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162 149 <span class="k">my</span><span class="s">(</span><span class="i">$RoundedValue</span><span class="s">)</span><span class="sc">;</span>
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163 150
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164 151 <span class="k">if</span> <span class="s">(</span><span class="k">defined</span><span class="s">(</span><span class="i">$DecimalPlaces</span><span class="s">)</span> && <span class="i">$DecimalPlaces</span> > <span class="n">0</span><span class="s">)</span> <span class="s">{</span>
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165 152 <span class="i">$RoundedValue</span> = <span class="k">sprintf</span> <span class="q">"%.${DecimalPlaces}f"</span><span class="cm">,</span> <span class="i">$Value</span><span class="sc">;</span>
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166 153 <span class="s">}</span>
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167 154 <span class="k">else</span> <span class="s">{</span>
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168 155 <span class="k">if</span> <span class="s">(</span><span class="i">$Value</span> < <span class="n">0</span><span class="s">)</span> <span class="s">{</span>
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169 156 <span class="i">$RoundedValue</span> = <span class="k">int</span><span class="s">(</span><span class="i">$Value</span> - <span class="n">0.5</span><span class="s">)</span><span class="sc">;</span>
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170 157 <span class="s">}</span>
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171 158 <span class="k">else</span> <span class="s">{</span>
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172 159 <span class="i">$RoundedValue</span> = <span class="k">int</span><span class="s">(</span><span class="i">$Value</span> + <span class="n">0.5</span><span class="s">)</span><span class="sc">;</span>
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173 160 <span class="s">}</span>
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174 161 <span class="s">}</span>
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175 162 <span class="k">return</span> <span class="i">$RoundedValue</span><span class="sc">;</span>
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176 163 <span class="s">}</span>
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177 164
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178 165 <span class="c"># Return tangent of an angle expressed in radians.</span>
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179 <a name="tan-"></a> 166 <span class="k">sub </span><span class="m">tan</span> <span class="s">{</span>
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180 167 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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181 168
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182 169 <span class="k">return</span> <span class="s">(</span><span class="i">CORE::</span><span class="k">sin</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span>/<span class="i">CORE::</span><span class="k">cos</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span><span class="s">)</span><span class="sc">;</span>
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183 170 <span class="s">}</span>
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184 171
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185 172 <span class="c"># Return inverse sine of an angle expressed in radians.</span>
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186 173 <span class="c">#</span>
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187 174 <span class="c"># For a right angle triangle defined by sides X and Y in a unit circle, Pythagorean theorem implies</span>
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188 175 <span class="c"># X**2 + Y**2 = 1 and sin value corresponds to Y. So asin is equivalent to atan2(Y, sqrt(1-Y**2)).</span>
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189 176 <span class="c"># However, taking sqrt of negative numbers is problematic; Math::Trig::asin handles it using complex</span>
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190 177 <span class="c"># numbers.</span>
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191 178 <span class="c">#</span>
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192 <a name="asin-"></a> 179 <span class="k">sub </span><span class="m">asin ($)</span> <span class="s">{</span>
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193 180 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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194 181
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195 182 <span class="k">return</span> <span class="i">Math::Trig::asin</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span><span class="sc">;</span>
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196 183 <span class="s">}</span>
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197 184
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198 185 <span class="c"># Return inverse cosine of an angle expressed in radians.</span>
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199 186 <span class="c">#</span>
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200 187 <span class="c"># For a right angle triangle defined by sides X and Y in a unit circle, Pythagorean theorem implies</span>
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201 188 <span class="c"># X**2 + Y**2 = 1 and cos value corresponds to X. So asin is equivalent to atan2(sqrt(1-X**2), X)</span>
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202 189 <span class="c"># However, taking sqrt of negative numbers is problematic; Math::Trig::acos handles it using complex</span>
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203 190 <span class="c"># numbers.</span>
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204 191 <span class="c">#</span>
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205 <a name="acos-"></a> 192 <span class="k">sub </span><span class="m">acos ($)</span> <span class="s">{</span>
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206 193 <span class="k">my</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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207 194
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208 195 <span class="k">return</span> <span class="i">Math::Trig::acos</span><span class="s">(</span><span class="i">$Value</span><span class="s">)</span><span class="sc">;</span>
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209 196 <span class="s">}</span>
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210 197
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211 198 <span class="c"># Generate prime numbers up to a specified limit and return a reference to an</span>
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212 199 <span class="c"># array containing the prime numbers.</span>
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213 200 <span class="c">#</span>
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214 201 <span class="c"># By default, the first 1000 prime numbers are generated. The 1000th prime</span>
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215 202 <span class="c"># number is 7919 and that's why default limit is set to 7920.</span>
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216 203 <span class="c">#</span>
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217 <a name="GeneratePrimeNumbersUpToLimit-"></a> 204 <span class="k">sub </span><span class="m">GeneratePrimeNumbersUpToLimit (;$)</span> <span class="s">{</span>
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218 205 <span class="k">my</span><span class="s">(</span><span class="i">$Limit</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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219 206
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220 207 <span class="i">$Limit</span> = <span class="k">defined</span> <span class="i">$Limit</span> ? <span class="i">$Limit</span> <span class="co">:</span> <span class="n">7920</span><span class="sc">;</span>
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221 208
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222 209 <span class="k">return</span> <span class="i">_GeneratePrimeNumbers</span><span class="s">(</span><span class="q">'ByLimit'</span><span class="cm">,</span> <span class="i">$Limit</span><span class="s">)</span>
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223 210 <span class="s">}</span>
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224 211
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225 212 <span class="c"># Generate prime numbers up to specified count of prime numbers and return a</span>
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226 213 <span class="c"># reference to an array containing the prime numbers.</span>
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227 214 <span class="c">#</span>
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228 215 <span class="c"># By default, the first 1000 prime numbers are generated. The 1000th prime</span>
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229 216 <span class="c"># number is 7919.</span>
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230 217 <span class="c">#</span>
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231 <a name="GeneratePrimeNumbersUpToCount-"></a> 218 <span class="k">sub </span><span class="m">GeneratePrimeNumbersUpToCount (;$)</span> <span class="s">{</span>
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232 219 <span class="k">my</span><span class="s">(</span><span class="i">$Count</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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233 220
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234 221 <span class="i">$Count</span> = <span class="k">defined</span> <span class="i">$Count</span> ? <span class="i">$Count</span> <span class="co">:</span> <span class="n">1000</span><span class="sc">;</span>
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235 222
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236 223 <span class="k">return</span> <span class="i">_GeneratePrimeNumbers</span><span class="s">(</span><span class="q">'ByCount'</span><span class="cm">,</span> <span class="i">$Count</span><span class="s">)</span>
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237 224 <span class="s">}</span>
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238 225
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239 226 <span class="c"># Generate prime numbers up to specified limit or count and return a reference</span>
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240 227 <span class="c"># to an array containing the prime numbers.</span>
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241 228 <span class="c">#</span>
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242 229 <span class="c"># The algorithm to generate prime numbers is a modification of Sieve of Erastothenes</span>
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243 230 <span class="c"># prime number generator.</span>
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244 231 <span class="c">#</span>
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245 <a name="_GeneratePrimeNumbers-"></a> 232 <span class="k">sub </span><span class="m">_GeneratePrimeNumbers</span> <span class="s">{</span>
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246 233 <span class="k">my</span><span class="s">(</span><span class="i">$Mode</span><span class="cm">,</span> <span class="i">$Value</span><span class="s">)</span> = <span class="i">@_</span><span class="sc">;</span>
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247 234 <span class="k">my</span><span class="s">(</span><span class="i">$ByLimit</span><span class="cm">,</span> <span class="i">$PrimeNumber</span><span class="cm">,</span> <span class="i">$Number</span><span class="cm">,</span> <span class="i">$SqrtOfNumber</span><span class="cm">,</span> <span class="i">$NumberIsPrime</span><span class="cm">,</span> <span class="i">@PrimeNumbers</span><span class="s">)</span><span class="sc">;</span>
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248 235
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249 236 <span class="i">$ByLimit</span> = <span class="s">(</span><span class="i">$Mode</span> =~ <span class="q">/^ByLimit$/i</span><span class="s">)</span> ? <span class="n">1</span> <span class="co">:</span> <span class="n">0</span><span class="sc">;</span>
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250 237
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251 238 <span class="i">@PrimeNumbers</span> = <span class="s">(</span><span class="n">2</span><span class="cm">,</span> <span class="n">3</span><span class="s">)</span><span class="sc">;</span>
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252 239 <span class="i">$Number</span> = <span class="n">3</span><span class="sc">;</span>
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253 240
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254 241 <span class="c"># while ($Number <= $Limit) {</span>
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255 242 <span class="k">while</span> <span class="s">(</span><span class="i">$ByLimit</span> ? <span class="s">(</span><span class="i">$Number</span> < <span class="i">$Value</span><span class="s">)</span> <span class="co">:</span> <span class="s">(</span><span class="i">@PrimeNumbers</span> < <span class="i">$Value</span><span class="s">)</span><span class="s">)</span> <span class="s">{</span>
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256 243 <span class="i">$Number</span> += <span class="n">2</span><span class="sc">;</span>
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257 244 <span class="i">$SqrtOfNumber</span> = <span class="k">sqrt</span> <span class="i">$Number</span><span class="sc">;</span>
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258 245
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259 246 <span class="i">$NumberIsPrime</span> = <span class="n">1</span><span class="sc">;</span>
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260 247 <span class="j">PRIMENUMBER:</span> <span class="k">for</span> <span class="i">$PrimeNumber</span> <span class="s">(</span><span class="i">@PrimeNumbers</span><span class="s">)</span> <span class="s">{</span>
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261 248 <span class="k">if</span> <span class="s">(</span><span class="i">$PrimeNumber</span> > <span class="i">$SqrtOfNumber</span><span class="s">)</span> <span class="s">{</span>
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262 249 <span class="k">last</span> <span class="j">PRIMENUMBER</span><span class="sc">;</span>
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263 250 <span class="s">}</span>
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264 251 <span class="k">if</span> <span class="s">(</span>!<span class="s">(</span><span class="i">$Number</span> % <span class="i">$PrimeNumber</span><span class="s">)</span><span class="s">)</span> <span class="s">{</span>
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265 252 <span class="i">$NumberIsPrime</span> = <span class="n">0</span><span class="sc">;</span>
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266 253 <span class="k">last</span> <span class="j">PRIMENUMBER</span><span class="sc">;</span>
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267 254 <span class="s">}</span>
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268 255 <span class="s">}</span>
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269 256 <span class="k">if</span> <span class="s">(</span><span class="i">$NumberIsPrime</span><span class="s">)</span> <span class="s">{</span>
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270 257 <span class="k">push</span> <span class="i">@PrimeNumbers</span><span class="cm">,</span> <span class="i">$Number</span><span class="sc">;</span>
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271 258 <span class="s">}</span>
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272 259 <span class="s">}</span>
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273 260 <span class="k">return</span> \<span class="i">@PrimeNumbers</span><span class="sc">;</span>
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274 261 <span class="s">}</span>
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275 262
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276 <a name="EOF-"></a></pre>
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277 <p> </p>
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278 <br />
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279 <center>
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280 <img src="../../../images/h2o2.png">
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281 </center>
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282 </body>
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283 </html>
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