/usr/share/gap/doc/ref/chap15.html is in gap-doc 4r7p5-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 | <?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 15: Number Theory</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap15" onload="jscontent()">
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap14.html">[Previous Chapter]</a> <a href="chap16.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap15_mj.html">[MathJax on]</a></p>
<p><a id="X7FB995737B7ED8A2" name="X7FB995737B7ED8A2"></a></p>
<div class="ChapSects"><a href="chap15.html#X7FB995737B7ED8A2">15 <span class="Heading">Number Theory</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7845C1F97A1742C7">15.1 <span class="Heading">InfoNumtheor (Info Class)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X796F0DFE7D5D211C">15.1-1 InfoNumtheor</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X823386567DAC22E6">15.2 <span class="Heading">Prime Residues</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X7FA3F5347B7004BA">15.2-1 PrimeResidues</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X85A0C67982D9057A">15.2-2 Phi</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X85296F3087611B03">15.2-3 Lambda</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X7D191CF67E5018BE">15.2-4 GeneratorsPrimeResidues</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X83103A5385821BAE">15.3 <span class="Heading">Primitive Roots and Discrete Logarithms</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X82373F3D8277EE9E">15.3-1 OrderMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X81AD9C7779A7BA89">15.3-2 LogMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X82440BB9812FF148">15.3-3 PrimitiveRootMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X790466C07BD90E20">15.3-4 IsPrimitiveRootMod</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7F9069D77AC48054">15.4 <span class="Heading">Roots Modulo Integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X83449DBC80495971">15.4-1 Jacobi</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X81464ABF7F10E544">15.4-2 Legendre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X83E3ED577B7A04ED">15.4-3 RootMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X84D3F03B862841F8">15.4-4 RootsMod</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X81F856E682A8ECBA">15.4-5 RootsUnityMod</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7B3A5A0378A32F83">15.5 <span class="Heading">Multiplicative Arithmetic Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X823707DF821E79A0">15.5-1 Sigma</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X798C62847EE0372E">15.5-2 Tau</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X79C1DA36827C2959">15.5-3 MoebiusMu</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7B2E061C835159B9">15.6 <span class="Heading">Continued Fractions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X874C161B83416092">15.6-1 ContinuedFractionExpansionOfRoot</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X8059667580A039A6">15.6-2 ContinuedFractionApproximationOfRoot</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap15.html#X7C5563A37D566DA5">15.7 <span class="Heading">Miscellaneous</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap15.html#X85E1EFC484F648A4">15.7-1 TwoSquares</a></span>
</div></div>
</div>
<h3>15 <span class="Heading">Number Theory</span></h3>
<p><strong class="pkg">GAP</strong> provides a couple of elementary number theoretic functions. Most of these deal with the group of integers coprime to <span class="SimpleMath">m</span>, called the <em>prime residue group</em>. The order of this group is <span class="SimpleMath">ϕ(m)</span> (see <code class="func">Phi</code> (<a href="chap15.html#X85A0C67982D9057A"><span class="RefLink">15.2-2</span></a>)), and <span class="SimpleMath">λ(m)</span> (see <code class="func">Lambda</code> (<a href="chap15.html#X85296F3087611B03"><span class="RefLink">15.2-3</span></a>)) is its exponent. This group is cyclic if and only if <span class="SimpleMath">m</span> is 2, 4, an odd prime power <span class="SimpleMath">p^n</span>, or twice an odd prime power <span class="SimpleMath">2 p^n</span>. In this case the generators of the group, i.e., elements of order <span class="SimpleMath">ϕ(m)</span>, are called <em>primitive roots</em> (see <code class="func">PrimitiveRootMod</code> (<a href="chap15.html#X82440BB9812FF148"><span class="RefLink">15.3-3</span></a>)).</p>
<p>Note that neither the arguments nor the return values of the functions listed below are groups or group elements in the sense of <strong class="pkg">GAP</strong>. The arguments are simply integers.</p>
<p><a id="X7845C1F97A1742C7" name="X7845C1F97A1742C7"></a></p>
<h4>15.1 <span class="Heading">InfoNumtheor (Info Class)</span></h4>
<p><a id="X796F0DFE7D5D211C" name="X796F0DFE7D5D211C"></a></p>
<h5>15.1-1 InfoNumtheor</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InfoNumtheor</code></td><td class="tdright">( info class )</td></tr></table></div>
<p><code class="func">InfoNumtheor</code> is the info class (see <a href="chap7.html#X7A9C902479CB6F7C"><span class="RefLink">7.4</span></a>) for the functions in the number theory chapter.</p>
<p><a id="X823386567DAC22E6" name="X823386567DAC22E6"></a></p>
<h4>15.2 <span class="Heading">Prime Residues</span></h4>
<p><a id="X7FA3F5347B7004BA" name="X7FA3F5347B7004BA"></a></p>
<h5>15.2-1 PrimeResidues</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimeResidues</code>( <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">PrimeResidues</code> returns the set of integers from the range <code class="code">[ 0 .. Abs( <var class="Arg">m</var> )-1 ]</code> that are coprime to the integer <var class="Arg">m</var>.</p>
<p><code class="code">Abs(<var class="Arg">m</var>)</code> must be less than <span class="SimpleMath">2^28</span>, otherwise the set would probably be too large anyhow.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimeResidues( 0 ); PrimeResidues( 1 ); PrimeResidues( 20 );</span>
[ ]
[ 0 ]
[ 1, 3, 7, 9, 11, 13, 17, 19 ]
</pre></div>
<p><a id="X85A0C67982D9057A" name="X85A0C67982D9057A"></a></p>
<h5>15.2-2 Phi</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Phi</code>( <var class="Arg">m</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">Phi</code> returns the number <span class="SimpleMath">ϕ(<var class="Arg">m</var>)</span> of positive integers less than the positive integer <var class="Arg">m</var> that are coprime to <var class="Arg">m</var>.</p>
<p>Suppose that <span class="SimpleMath">m = p_1^{e_1} p_2^{e_2} ⋯ p_k^{e_k}</span>. Then <span class="SimpleMath">ϕ(m)</span> is <span class="SimpleMath">p_1^{e_1-1} (p_1-1) p_2^{e_2-1} (p_2-1) ⋯ p_k^{e_k-1} (p_k-1)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Phi( 12 );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Phi( 2^13-1 ); # this proves that 2^(13)-1 is a prime</span>
8190
<span class="GAPprompt">gap></span> <span class="GAPinput">Phi( 2^15-1 );</span>
27000
</pre></div>
<p><a id="X85296F3087611B03" name="X85296F3087611B03"></a></p>
<h5>15.2-3 Lambda</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Lambda</code>( <var class="Arg">m</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">Lambda</code> returns the exponent <span class="SimpleMath">λ(<var class="Arg">m</var>)</span> of the group of prime residues modulo the integer <var class="Arg">m</var>.</p>
<p><span class="SimpleMath">λ(<var class="Arg">m</var>)</span> is the smallest positive integer <span class="SimpleMath">l</span> such that for every <span class="SimpleMath">a</span> relatively prime to <var class="Arg">m</var> we have <span class="SimpleMath">a^l ≡ 1 mod <var class="Arg">m</var></span>. Fermat's theorem asserts <span class="SimpleMath">a^{ϕ(<var class="Arg">m</var>)} ≡ 1 mod <var class="Arg">m</var></span>; thus <span class="SimpleMath">λ(<var class="Arg">m</var>)</span> divides <span class="SimpleMath">ϕ(<var class="Arg">m</var>)</span> (see <code class="func">Phi</code> (<a href="chap15.html#X85A0C67982D9057A"><span class="RefLink">15.2-2</span></a>)).</p>
<p>Carmichael's theorem states that <span class="SimpleMath">λ</span> can be computed as follows: <span class="SimpleMath">λ(2) = 1</span>, <span class="SimpleMath">λ(4) = 2</span> and <span class="SimpleMath">λ(2^e) = 2^{e-2}</span> if <span class="SimpleMath">3 ≤ e</span>, <span class="SimpleMath">λ(p^e) = (p-1) p^{e-1}</span> (i.e. <span class="SimpleMath">ϕ(m)</span>) if <span class="SimpleMath">p</span> is an odd prime and <span class="SimpleMath">λ(m*n) =</span><code class="code">Lcm</code><span class="SimpleMath">( λ(m), λ(n) )</span> if <span class="SimpleMath">m, n</span> are coprime.</p>
<p>Composites for which <span class="SimpleMath">λ(m)</span> divides <span class="SimpleMath">m - 1</span> are called Carmichaels. If <span class="SimpleMath">6k+1</span>, <span class="SimpleMath">12k+1</span> and <span class="SimpleMath">18k+1</span> are primes their product is such a number. There are only 1547 Carmichaels below <span class="SimpleMath">10^10</span> but 455052511 primes.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Lambda( 10 );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Lambda( 30 );</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">Lambda( 561 ); # 561 is the smallest Carmichael number</span>
80
</pre></div>
<p><a id="X7D191CF67E5018BE" name="X7D191CF67E5018BE"></a></p>
<h5>15.2-4 GeneratorsPrimeResidues</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsPrimeResidues</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">n</var> be a positive integer. <code class="func">GeneratorsPrimeResidues</code> returns a description of generators of the group of prime residues modulo <var class="Arg">n</var>. The return value is a record with components</p>
<dl>
<dt><strong class="Mark"><code class="code">primes</code>: </strong></dt>
<dd><p>a list of the prime factors of <var class="Arg">n</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">exponents</code>: </strong></dt>
<dd><p>a list of the exponents of these primes in the factorization of <var class="Arg">n</var>, and</p>
</dd>
<dt><strong class="Mark"><code class="code">generators</code>: </strong></dt>
<dd><p>a list describing generators of the group of prime residues; for the prime factor <span class="SimpleMath">2</span>, either a primitive root or a list of two generators is stored, for each other prime factor of <var class="Arg">n</var>, a primitive root is stored.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsPrimeResidues( 1 );</span>
rec( exponents := [ ], generators := [ ], primes := [ ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsPrimeResidues( 4*3 );</span>
rec( exponents := [ 2, 1 ], generators := [ 7, 5 ],
primes := [ 2, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsPrimeResidues( 8*9*5 );</span>
rec( exponents := [ 3, 2, 1 ],
generators := [ [ 271, 181 ], 281, 217 ], primes := [ 2, 3, 5 ] )
</pre></div>
<p><a id="X83103A5385821BAE" name="X83103A5385821BAE"></a></p>
<h4>15.3 <span class="Heading">Primitive Roots and Discrete Logarithms</span></h4>
<p><a id="X82373F3D8277EE9E" name="X82373F3D8277EE9E"></a></p>
<h5>15.3-1 OrderMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderMod</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">OrderMod</code> returns the multiplicative order of the integer <var class="Arg">n</var> modulo the positive integer <var class="Arg">m</var>. If <var class="Arg">n</var> and <var class="Arg">m</var> are not coprime the order of <var class="Arg">n</var> is not defined and <code class="func">OrderMod</code> will return <code class="code">0</code>.</p>
<p>If <var class="Arg">n</var> and <var class="Arg">m</var> are relatively prime the multiplicative order of <var class="Arg">n</var> modulo <var class="Arg">m</var> is the smallest positive integer <span class="SimpleMath">i</span> such that <span class="SimpleMath"><var class="Arg">n</var>^i ≡ 1 mod <var class="Arg">m</var></span>. If the group of prime residues modulo <var class="Arg">m</var> is cyclic then each element of maximal order is called a primitive root modulo <var class="Arg">m</var> (see <code class="func">IsPrimitiveRootMod</code> (<a href="chap15.html#X790466C07BD90E20"><span class="RefLink">15.3-4</span></a>)).</p>
<p><code class="func">OrderMod</code> usually spends most of its time factoring <var class="Arg">m</var> and <span class="SimpleMath">ϕ(<var class="Arg">m</var>)</span> (see <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OrderMod( 2, 7 );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">OrderMod( 3, 7 ); # 3 is a primitive root modulo 7</span>
6
</pre></div>
<p><a id="X81AD9C7779A7BA89" name="X81AD9C7779A7BA89"></a></p>
<h5>15.3-2 LogMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LogMod</code>( <var class="Arg">n</var>, <var class="Arg">r</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LogModShanks</code>( <var class="Arg">n</var>, <var class="Arg">r</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes the discrete <var class="Arg">r</var>-logarithm of the integer <var class="Arg">n</var> modulo the integer <var class="Arg">m</var>. It returns a number <var class="Arg">l</var> such that <span class="SimpleMath"><var class="Arg">r</var>^<var class="Arg">l</var> ≡ <var class="Arg">n</var> mod <var class="Arg">m</var></span> if such a number exists. Otherwise <code class="keyw">fail</code> is returned.</p>
<p><code class="func">LogModShanks</code> uses the Baby Step - Giant Step Method of Shanks (see for example <a href="chapBib.html#biBCoh93">[Coh93, section 5.4.1]</a>) and in general requires more memory than a call to <code class="func">LogMod</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= LogMod( 2, 5, 7 ); 5^l mod 7 = 2;</span>
4
true
<span class="GAPprompt">gap></span> <span class="GAPinput">LogMod( 1, 3, 3 ); LogMod( 2, 3, 3 );</span>
0
fail
</pre></div>
<p><a id="X82440BB9812FF148" name="X82440BB9812FF148"></a></p>
<h5>15.3-3 PrimitiveRootMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveRootMod</code>( <var class="Arg">m</var>[, <var class="Arg">start</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">PrimitiveRootMod</code> returns the smallest primitive root modulo the positive integer <var class="Arg">m</var> and <code class="keyw">fail</code> if no such primitive root exists. If the optional second integer argument <var class="Arg">start</var> is given <code class="func">PrimitiveRootMod</code> returns the smallest primitive root that is strictly larger than <var class="Arg">start</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># largest primitive root for a prime less than 2000:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveRootMod( 409 ); </span>
21
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveRootMod( 541, 2 );</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput"># 327 is the largest primitive root mod 337:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveRootMod( 337, 327 );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput"># there exists no primitive root modulo 30:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimitiveRootMod( 30 );</span>
fail
</pre></div>
<p><a id="X790466C07BD90E20" name="X790466C07BD90E20"></a></p>
<h5>15.3-4 IsPrimitiveRootMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPrimitiveRootMod</code>( <var class="Arg">r</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IsPrimitiveRootMod</code> returns <code class="keyw">true</code> if the integer <var class="Arg">r</var> is a primitive root modulo the positive integer <var class="Arg">m</var>, and <code class="keyw">false</code> otherwise. If <var class="Arg">r</var> is less than 0 or larger than <var class="Arg">m</var> it is replaced by its remainder.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimitiveRootMod( 2, 541 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimitiveRootMod( -539, 541 ); # same computation as above;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPrimitiveRootMod( 4, 541 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAny( [1..29], r -> IsPrimitiveRootMod( r, 30 ) );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput"># there is no a primitive root modulo 30</span>
</pre></div>
<p><a id="X7F9069D77AC48054" name="X7F9069D77AC48054"></a></p>
<h4>15.4 <span class="Heading">Roots Modulo Integers</span></h4>
<p><a id="X83449DBC80495971" name="X83449DBC80495971"></a></p>
<h5>15.4-1 Jacobi</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Jacobi</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">Jacobi</code> returns the value of the <em>Kronecker-Jacobi symbol</em> <span class="SimpleMath">J(<var class="Arg">n</var>,<var class="Arg">m</var>)</span> of the integer <var class="Arg">n</var> modulo the integer <var class="Arg">m</var>. It is defined as follows:</p>
<p>If <span class="SimpleMath">n</span> and <span class="SimpleMath">m</span> are not coprime then <span class="SimpleMath">J(n,m) = 0</span>. Furthermore, <span class="SimpleMath">J(n,1) = 1</span> and <span class="SimpleMath">J(n,-1) = -1</span> if <span class="SimpleMath">m < 0</span> and <span class="SimpleMath">+1</span> otherwise. And for odd <span class="SimpleMath">n</span> it is <span class="SimpleMath">J(n,2) = (-1)^k</span> with <span class="SimpleMath">k = (n^2-1)/8</span>. For odd primes <span class="SimpleMath">m</span> which are coprime to <span class="SimpleMath">n</span> the Kronecker-Jacobi symbol has the same value as the Legendre symbol (see <code class="func">Legendre</code> (<a href="chap15.html#X81464ABF7F10E544"><span class="RefLink">15.4-2</span></a>)).</p>
<p>For the general case suppose that <span class="SimpleMath">m = p_1 ⋅ p_2 ⋯ p_k</span> is a product of <span class="SimpleMath">-1</span> and of primes, not necessarily distinct, and that <span class="SimpleMath">n</span> is coprime to <span class="SimpleMath">m</span>. Then <span class="SimpleMath">J(n,m) = J(n,p_1) ⋅ J(n,p_2) ⋯ J(n,p_k)</span>.</p>
<p>Note that the Kronecker-Jacobi symbol coincides with the Jacobi symbol that is defined for odd <span class="SimpleMath">m</span> in many number theory books. For odd primes <span class="SimpleMath">m</span> and <span class="SimpleMath">n</span> coprime to <span class="SimpleMath">m</span> it coincides with the Legendre symbol.</p>
<p><code class="func">Jacobi</code> is very efficient, even for large values of <var class="Arg">n</var> and <var class="Arg">m</var>, it is about as fast as the Euclidean algorithm (see <code class="func">Gcd</code> (<a href="chap56.html#X7DE207718456F98F"><span class="RefLink">56.7-1</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Jacobi( 11, 35 ); # 9^2 = 11 mod 35</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput"># this is -1, thus there is no r such that r^2 = 6 mod 35</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Jacobi( 6, 35 );</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput"># this is 1 even though there is no r with r^2 = 3 mod 35</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Jacobi( 3, 35 );</span>
1
</pre></div>
<p><a id="X81464ABF7F10E544" name="X81464ABF7F10E544"></a></p>
<h5>15.4-2 Legendre</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Legendre</code>( <var class="Arg">n</var>, <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">Legendre</code> returns the value of the <em>Legendre symbol</em> of the integer <var class="Arg">n</var> modulo the positive integer <var class="Arg">m</var>.</p>
<p>The value of the Legendre symbol <span class="SimpleMath">L(n/m)</span> is 1 if <span class="SimpleMath">n</span> is a <em>quadratic residue</em> modulo <span class="SimpleMath">m</span>, i.e., if there exists an integer <span class="SimpleMath">r</span> such that <span class="SimpleMath">r^2 ≡ n mod m</span> and <span class="SimpleMath">-1</span> otherwise.</p>
<p>If a root of <var class="Arg">n</var> exists it can be found by <code class="func">RootMod</code> (<a href="chap15.html#X83E3ED577B7A04ED"><span class="RefLink">15.4-3</span></a>).</p>
<p>While the value of the Legendre symbol usually is only defined for <var class="Arg">m</var> a prime, we have extended the definition to include composite moduli too. The Jacobi symbol (see <code class="func">Jacobi</code> (<a href="chap15.html#X83449DBC80495971"><span class="RefLink">15.4-1</span></a>)) is another generalization of the Legendre symbol for composite moduli that is much cheaper to compute, because it does not need the factorization of <var class="Arg">m</var> (see <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>)).</p>
<p>A description of the Jacobi symbol, the Legendre symbol, and related topics can be found in <a href="chapBib.html#biBBaker84">[Bak84]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Legendre( 5, 11 ); # 4^2 = 5 mod 11</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput"># this is -1, thus there is no r such that r^2 = 6 mod 11</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Legendre( 6, 11 );</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput"># this is -1, thus there is no r such that r^2 = 3 mod 35</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Legendre( 3, 35 );</span>
-1
</pre></div>
<p><a id="X83E3ED577B7A04ED" name="X83E3ED577B7A04ED"></a></p>
<h5>15.4-3 RootMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RootMod</code>( <var class="Arg">n</var>[, <var class="Arg">k</var>], <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">RootMod</code> computes a <var class="Arg">k</var>th root of the integer <var class="Arg">n</var> modulo the positive integer <var class="Arg">m</var>, i.e., a <span class="SimpleMath">r</span> such that <span class="SimpleMath">r^<var class="Arg">k</var> ≡ <var class="Arg">n</var> mod <var class="Arg">m</var></span>. If no such root exists <code class="func">RootMod</code> returns <code class="keyw">fail</code>. If only the arguments <var class="Arg">n</var> and <var class="Arg">m</var> are given, the default value for <var class="Arg">k</var> is <span class="SimpleMath">2</span>.</p>
<p>A square root of <var class="Arg">n</var> exists only if <code class="code">Legendre(<var class="Arg">n</var>,<var class="Arg">m</var>) = 1</code> (see <code class="func">Legendre</code> (<a href="chap15.html#X81464ABF7F10E544"><span class="RefLink">15.4-2</span></a>)). If <var class="Arg">m</var> has <span class="SimpleMath">r</span> different prime factors then there are <span class="SimpleMath">2^r</span> different roots of <var class="Arg">n</var> mod <var class="Arg">m</var>. It is unspecified which one <code class="func">RootMod</code> returns. You can, however, use <code class="func">RootsMod</code> (<a href="chap15.html#X84D3F03B862841F8"><span class="RefLink">15.4-4</span></a>) to compute the full set of roots.</p>
<p><code class="func">RootMod</code> is efficient even for large values of <var class="Arg">m</var>, in fact the most time is usually spent factoring <var class="Arg">m</var> (see <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># note 'RootMod' does not return 8 in this case but -8:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RootMod( 64, 1009 );</span>
1001
<span class="GAPprompt">gap></span> <span class="GAPinput">RootMod( 64, 3, 1009 );</span>
518
<span class="GAPprompt">gap></span> <span class="GAPinput">RootMod( 64, 5, 1009 );</span>
656
<span class="GAPprompt">gap></span> <span class="GAPinput">List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> x mod 1009 ); # set of all square roots of 64 mod 1009</span>
[ 1001, 8 ]
</pre></div>
<p><a id="X84D3F03B862841F8" name="X84D3F03B862841F8"></a></p>
<h5>15.4-4 RootsMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RootsMod</code>( <var class="Arg">n</var>[, <var class="Arg">k</var>], <var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">RootsMod</code> computes the set of <var class="Arg">k</var>th roots of the integer <var class="Arg">n</var> modulo the positive integer <var class="Arg">m</var>, i.e., the list of all <span class="SimpleMath">r</span> such that <span class="SimpleMath">r^<var class="Arg">k</var> ≡ <var class="Arg">n</var> mod <var class="Arg">m</var></span>. If only the arguments <var class="Arg">n</var> and <var class="Arg">m</var> are given, the default value for <var class="Arg">k</var> is <span class="SimpleMath">2</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RootsMod( 1, 7*31 ); # the same as `RootsUnityMod( 7*31 )'</span>
[ 1, 92, 125, 216 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RootsMod( 7, 7*31 );</span>
[ 21, 196 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RootsMod( 5, 7*31 );</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RootsMod( 1, 5, 7*31 );</span>
[ 1, 8, 64, 78, 190 ]
</pre></div>
<p><a id="X81F856E682A8ECBA" name="X81F856E682A8ECBA"></a></p>
<h5>15.4-5 RootsUnityMod</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RootsUnityMod</code>( [<var class="Arg">k</var>, ]<var class="Arg">m</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">RootsUnityMod</code> returns the set of <var class="Arg">k</var>-th roots of unity modulo the positive integer <var class="Arg">m</var>, i.e., the list of all solutions <span class="SimpleMath">r</span> of <span class="SimpleMath">r^<var class="Arg">k</var> ≡ <var class="Arg">n</var> mod <var class="Arg">m</var></span>. If only the argument <var class="Arg">m</var> is given, the default value for <var class="Arg">k</var> is <span class="SimpleMath">2</span>.</p>
<p>In general there are <span class="SimpleMath"><var class="Arg">k</var>^n</span> such roots if the modulus <var class="Arg">m</var> has <span class="SimpleMath">n</span> different prime factors <span class="SimpleMath">p</span> such that <span class="SimpleMath">p ≡ 1 mod <var class="Arg">k</var></span>. If <span class="SimpleMath"><var class="Arg">k</var>^2</span> divides <var class="Arg">m</var> then there are <span class="SimpleMath"><var class="Arg">k</var>^{n+1}</span> such roots; and especially if <span class="SimpleMath"><var class="Arg">k</var> = 2</span> and 8 divides <var class="Arg">m</var> there are <span class="SimpleMath">2^{n+2}</span> such roots.</p>
<p>In the current implementation <var class="Arg">k</var> must be a prime.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RootsUnityMod( 7*31 ); RootsUnityMod( 3, 7*31 );</span>
[ 1, 92, 125, 216 ]
[ 1, 25, 32, 36, 67, 149, 156, 191, 211 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">RootsUnityMod( 5, 7*31 );</span>
[ 1, 8, 64, 78, 190 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> x -> x mod 1009 ); # set of all square roots of 64 mod 1009</span>
[ 1001, 8 ]
</pre></div>
<p><a id="X7B3A5A0378A32F83" name="X7B3A5A0378A32F83"></a></p>
<h4>15.5 <span class="Heading">Multiplicative Arithmetic Functions</span></h4>
<p><a id="X823707DF821E79A0" name="X823707DF821E79A0"></a></p>
<h5>15.5-1 Sigma</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Sigma</code>( <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">Sigma</code> returns the sum of the positive divisors of the nonzero integer <var class="Arg">n</var>.</p>
<p><code class="func">Sigma</code> is a multiplicative arithmetic function, i.e., if <span class="SimpleMath">n</span> and <span class="SimpleMath">m</span> are relatively prime we have that <span class="SimpleMath">σ(n ⋅ m) = σ(n) σ(m)</span>.</p>
<p>Together with the formula <span class="SimpleMath">σ(p^k) = (p^{k+1}-1) / (p-1)</span> this allows us to compute <span class="SimpleMath">σ(<var class="Arg">n</var>)</span>.</p>
<p>Integers <var class="Arg">n</var> for which <span class="SimpleMath">σ(<var class="Arg">n</var>) = 2 <var class="Arg">n</var></span> are called perfect. Even perfect integers are exactly of the form <span class="SimpleMath">2^{<var class="Arg">n</var>-1}(2^<var class="Arg">n</var>-1)</span> where <span class="SimpleMath">2^<var class="Arg">n</var>-1</span> is prime. Primes of the form <span class="SimpleMath">2^<var class="Arg">n</var>-1</span> are called <em>Mersenne primes</em>, and 42 among the known Mersenne primes are obtained for <var class="Arg">n</var> <span class="SimpleMath">=</span> 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583 and 25964951. Please find more up to date information about Mersenne primes at <span class="URL"><a href="http://www.mersenne.org">http://www.mersenne.org</a></span>. It is not known whether odd perfect integers exist, however <a href="chapBib.html#biBBC89">[BC89]</a> show that any such integer must have at least 300 decimal digits.</p>
<p><code class="func">Sigma</code> usually spends most of its time factoring <var class="Arg">n</var> (see <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Sigma( 1 );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Sigma( 1009 ); # 1009 is a prime</span>
1010
<span class="GAPprompt">gap></span> <span class="GAPinput">Sigma( 8128 ) = 2*8128; # 8128 is a perfect number</span>
true
</pre></div>
<p><a id="X798C62847EE0372E" name="X798C62847EE0372E"></a></p>
<h5>15.5-2 Tau</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tau</code>( <var class="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">Tau</code> returns the number of the positive divisors of the nonzero integer <var class="Arg">n</var>.</p>
<p><code class="func">Tau</code> is a multiplicative arithmetic function, i.e., if <span class="SimpleMath">n</span> and <span class="SimpleMath">m</span> are relative prime we have <span class="SimpleMath">τ(n ⋅ m) = τ(n) τ(m)</span>. Together with the formula <span class="SimpleMath">τ(p^k) = k+1</span> this allows us to compute <span class="SimpleMath">τ(<var class="Arg">n</var>)</span>.</p>
<p><code class="func">Tau</code> usually spends most of its time factoring <var class="Arg">n</var> (see <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Tau( 1 );</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">Tau( 1013 ); # thus 1013 is a prime</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">Tau( 8128 );</span>
14
<span class="GAPprompt">gap></span> <span class="GAPinput"># result is odd if and only if argument is a perfect square:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Tau( 36 );</span>
9
</pre></div>
<p><a id="X79C1DA36827C2959" name="X79C1DA36827C2959"></a></p>
<h5>15.5-3 MoebiusMu</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MoebiusMu</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">MoebiusMu</code> computes the value of Moebius inversion function for the nonzero integer <var class="Arg">n</var>. This is 0 for integers which are not squarefree, i.e., which are divided by a square <span class="SimpleMath">r^2</span>. Otherwise it is 1 if <var class="Arg">n</var> has a even number and <span class="SimpleMath">-1</span> if <var class="Arg">n</var> has an odd number of prime factors.</p>
<p>The importance of <span class="SimpleMath">μ</span> stems from the so called inversion formula. Suppose <span class="SimpleMath">f</span> is a multiplicative arithmetic function defined on the positive integers and let <span class="SimpleMath">g(n) = ∑_{d ∣ n} f(d)</span>. Then <span class="SimpleMath">f(n) = ∑_{d ∣ n} μ(d) g(n/d)</span>. As a special case we have <span class="SimpleMath">ϕ(n) = ∑_{d ∣ n} μ(d) n/d</span> since <span class="SimpleMath">n = ∑_{d ∣ n} ϕ(d)</span> (see <code class="func">Phi</code> (<a href="chap15.html#X85A0C67982D9057A"><span class="RefLink">15.2-2</span></a>)).</p>
<p><code class="func">MoebiusMu</code> usually spends all of its time factoring <var class="Arg">n</var> (see <code class="func">FactorsInt</code> (<a href="chap14.html#X82C989DB84744B36"><span class="RefLink">14.4-7</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">MoebiusMu( 60 ); MoebiusMu( 61 ); MoebiusMu( 62 );</span>
0
-1
1
</pre></div>
<p><a id="X7B2E061C835159B9" name="X7B2E061C835159B9"></a></p>
<h4>15.6 <span class="Heading">Continued Fractions</span></h4>
<p><a id="X874C161B83416092" name="X874C161B83416092"></a></p>
<h5>15.6-1 ContinuedFractionExpansionOfRoot</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContinuedFractionExpansionOfRoot</code>( <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first <var class="Arg">n</var> terms of the continued fraction expansion of the only positive real root of the polynomial <var class="Arg">f</var> with integer coefficients. The leading coefficient of <var class="Arg">f</var> must be positive and the value of <var class="Arg">f</var> at 0 must be negative. If the degree of <var class="Arg">f</var> is 2 and <var class="Arg">n</var> = 0, the function computes one period of the continued fraction expansion of the root in question. Anything may happen if <var class="Arg">f</var> has three or more positive real roots.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Indeterminate(Integers);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ContinuedFractionExpansionOfRoot(x^2-7,20);</span>
[ 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ContinuedFractionExpansionOfRoot(x^2-7,0);</span>
[ 2, 1, 1, 1, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ContinuedFractionExpansionOfRoot(x^3-2,20);</span>
[ 1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ContinuedFractionExpansionOfRoot(x^5-x-1,50);</span>
[ 1, 5, 1, 42, 1, 3, 24, 2, 2, 1, 16, 1, 11, 1, 1, 2, 31, 1, 12, 5,
1, 7, 11, 1, 4, 1, 4, 2, 2, 3, 4, 2, 1, 1, 11, 1, 41, 12, 1, 8, 1,
1, 1, 1, 1, 9, 2, 1, 5, 4 ]
</pre></div>
<p><a id="X8059667580A039A6" name="X8059667580A039A6"></a></p>
<h5>15.6-2 ContinuedFractionApproximationOfRoot</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ContinuedFractionApproximationOfRoot</code>( <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The <var class="Arg">n</var>th continued fraction approximation of the only positive real root of the polynomial <var class="Arg">f</var> with integer coefficients. The leading coefficient of <var class="Arg">f</var> must be positive and the value of <var class="Arg">f</var> at 0 must be negative. Anything may happen if <var class="Arg">f</var> has three or more positive real roots.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ContinuedFractionApproximationOfRoot(x^2-2,10);</span>
3363/2378
<span class="GAPprompt">gap></span> <span class="GAPinput">3363^2-2*2378^2;</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">z := ContinuedFractionApproximationOfRoot(x^5-x-1,20);</span>
499898783527/428250732317
<span class="GAPprompt">gap></span> <span class="GAPinput">z^5-z-1;</span>
486192462527432755459620441970617283/
14404247382319842421697357558805709031116987826242631261357
</pre></div>
<p><a id="X7C5563A37D566DA5" name="X7C5563A37D566DA5"></a></p>
<h4>15.7 <span class="Heading">Miscellaneous</span></h4>
<p><a id="X85E1EFC484F648A4" name="X85E1EFC484F648A4"></a></p>
<h5>15.7-1 TwoSquares</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwoSquares</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">TwoSquares</code> returns a list of two integers <span class="SimpleMath">x ≤ y</span> such that the sum of the squares of <span class="SimpleMath">x</span> and <span class="SimpleMath">y</span> is equal to the nonnegative integer <var class="Arg">n</var>, i.e., <span class="SimpleMath">n = x^2 + y^2</span>. If no such representation exists <code class="func">TwoSquares</code> will return <code class="keyw">fail</code>. <code class="func">TwoSquares</code> will return a representation for which the gcd of <span class="SimpleMath">x</span> and <span class="SimpleMath">y</span> is as small as possible. It is not specified which representation <code class="func">TwoSquares</code> returns if there is more than one.</p>
<p>Let <span class="SimpleMath">a</span> be the product of all maximal powers of primes of the form <span class="SimpleMath">4k+3</span> dividing <var class="Arg">n</var>. A representation of <var class="Arg">n</var> as a sum of two squares exists if and only if <span class="SimpleMath">a</span> is a perfect square. Let <span class="SimpleMath">b</span> be the maximal power of <span class="SimpleMath">2</span> dividing <var class="Arg">n</var> or its half, whichever is a perfect square. Then the minimal possible gcd of <span class="SimpleMath">x</span> and <span class="SimpleMath">y</span> is the square root <span class="SimpleMath">c</span> of <span class="SimpleMath">a ⋅ b</span>. The number of different minimal representation with <span class="SimpleMath">x ≤ y</span> is <span class="SimpleMath">2^{l-1}</span>, where <span class="SimpleMath">l</span> is the number of different prime factors of the form <span class="SimpleMath">4k+1</span> of <var class="Arg">n</var>.</p>
<p>The algorithm first finds a square root <span class="SimpleMath">r</span> of <span class="SimpleMath">-1</span> modulo <span class="SimpleMath"><var class="Arg">n</var> / (a ⋅ b)</span>, which must exist, and applies the Euclidean algorithm to <span class="SimpleMath">r</span> and <var class="Arg">n</var>. The first residues in the sequence that are smaller than <span class="SimpleMath">sqrt{<var class="Arg">n</var>/(a ⋅ b)}</span> times <span class="SimpleMath">c</span> are a possible pair <span class="SimpleMath">x</span> and <span class="SimpleMath">y</span>.</p>
<p>Better descriptions of the algorithm and related topics can be found in <a href="chapBib.html#biBWagon90">[Wag90]</a> and <a href="chapBib.html#biBZagier90">[Zag90]</a>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">TwoSquares( 5 );</span>
[ 1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TwoSquares( 11 ); # there is no representation</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">TwoSquares( 16 );</span>
[ 0, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># 3 is the minimal possible gcd because 9 divides 45:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TwoSquares( 45 );</span>
[ 3, 6 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># it is not [5,10] because their gcd is not minimal:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TwoSquares( 125 );</span>
[ 2, 11 ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># [10,11] would be the other possible representation:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TwoSquares( 13*17 );</span>
[ 5, 14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">TwoSquares( 848654483879497562821 ); # argument is prime</span>
[ 6305894639, 28440994650 ]
</pre></div>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap14.html">[Previous Chapter]</a> <a href="chap16.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|