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Fs（+1）1226年1777年Fv（，）k（或）h（the）g（con）m（tin）m（分数）f（膨胀）h（of）f Ft（x）h Fv（is）f（\014nite）g（and）hFt（n）g Fv（is）162 1897 y（大于）e（大于）i（分母）g h的e（有理）d（n）m（um）m（b）s（er）j Ft（x）p Fv（.）44 b（In）33b（the）g（\014rst）g（case，）1154 2024 y Fn（\014）1154 2084年（2014年）1154 2144年（2014）1154 2204年（014年）1188 2169年Ft（x）22 b Fp（\000）1375 2102 y Ft（p）1424 2117 y Fm（m）p1375 2146 116 4 v 1378 2237 a Ft（q）1421 2252 y Fm（m）15002024年Fn（2014）1500 2084年（2014）150 2144年（014）15002204年（\014）1561 2169年Ft（<）1841 2102年Fv（1）第1675页2146 382 4 v 1675 2237 a Ft（q）1718 2252 y Fm（m）1807 2237y Fp（\001）f Ft（q）1899 2252 y Fm（m）p Fs（+1）2094 2169 yFp（024）2427 2102 y Fv（1）p 2209 2146 485 4 v 2209 2237a Ft（q）2252 2252 y Fm（m）2341 2237 y Fp（\001）h Fv（）pFt（n）g Fv（+）g（1））2704 2169 y Ft（；）162 2482 y Fv（so）2902414 y Ft（a）p 290 2459 52 4 v 295 2550 a（b）379 2482 yFv（=）493 2414 y Ft（p）542 2429 y Fm（m）p 493 2459 116 4v 496 2550 a Ft（q）539 2565 y Fm（m）650 2482 y Fv（满足\014es）32b（f（结论）g（of）g（引理）41 b（In）31 b（第二）h（案例）f（just）g（let）3449 2414 y Ft（a）p 3449 2459 52 4v 3454 2550 a（b）3537 2482 y Fv（=）c Ft（x）p Fv（.）p 36532644 4 66 v 3657 2581 59 v 3657 2644 v 3715 264466 v 162 2847 a Fr（De\014nition）36 b（1.4.）42 b Fv（a）32b（代表）m（站）i Ft（n）28 b Fv（=）f Ft（x）1888 2811y Fs（2）1950年2847 y Fv（+）22 b Ft（y）2100年2811 y F（2）21712847 y Fv（is）32 b Fq（基本）g Fv（if）g（gcd）p Ft（x；）17b（y）t Fv（\））27 b（=）g（1.）162 3050 y Fr（引理）53 b（1.5.）48b Fq（If）f Ft（n）h Fq（is）f（可除）f（by）h（a）g（素数）gFt（p）g Fq（of）g（the）h（form）f Fv（4）p Ft（m）31 b Fv（+）g（3）pFq（，）50 b（then）e Ft（n）f Fq（has）g（no）162 3171 y（基本）34b（r）-5 b（epr）g（esentations.）162 3374 y（Pr）g（o）g（of）48b Fv（If）33 b Ft（n）g Fv（has）g（a）f（primitiv）m（e）e（represen）m（station）kFt（n）27 b Fv（=）h Ft（x）2237 3338 y Fs（2）2299 3374 y Fv（+）22b Ft（y）2449 3338 y Fs（2）2488 3374 y Fv（，）32 b（then）1217年3594 y Ft（p）c Fp（j）f Ft（x）1404 3553 y Fs（2）1466 3594y Fv（+）22 b Ft（y）1616 3553 y Fs（2）1785 3594 y Fv（和）147b（gcd）p Ft（x；）17 b（y）t Fv（\））27 b（=）g（1）p Ft（；）1623814 y Fv（so）33 b Ft（p）28 b Fo（-）f Ft（x）33 b Flv（and）gFt（p）28 b Fo（-）f Ft（y）t Fv（.）43 b（Th）m（us）34 b Ft（x）12483778 y Fs（2）1311 3814 y Fv（+）22 b Ft（y）1461 3778 y F（2）15273814 y Fp（\021）28 b Fv（0）44 b（\（mo）s（d）33 b Ft（p）p Fv（\））f（so）h（since）g Fo（Z）p Ft（=p）p Fo（Z）27 b Fv（is）33 b（a）f（\014eld）g（w）m（e）i（can）f（除以）162 3935 y（b）m（y）g Ft（y）349 3898 y Fs（2）420 3935y Fv（和）g（参见）h（即）1442 4055 y（）p Ft（x=y）t Fv（）16744014 y Fs（2）1740 4055 y Fp（\021）28 b（\000）p Fv（1）100b（\（mo）s（d）32 b Ft（p）p Fv（\））p Ft（：）162 4257 y Fv（Th）m（us）i（the）f（二次）f（剩余）h（sym）m（b）s（ol）1678 4146 y Fn（\020）17484218 y Fl（\000）p Fs（1）p 1748 4234 91 4 v 1775 4291 aFm（p）1848 4146 y Fn（\021）1940 4257 y Fv（等于）g（+1）43b（Ho）m（w）m（ev）m（er，）858 4433 y Fn（\022）942 4506 y Fp（\000）pFv（1）p 942 4551 127 4 v 980 4642 a Ft（p）1078 4433 y Fn（\023）11794574 y Fv（=）28 b（）p Fp（000）p Fv（1））1495 4502 y Fk（p）pFj（\000）p Fi（1）p 1495 4518 111 3 v 1535 4559 a（2）16474574年Fv（=）f（）p Fp（000）p Fv（1））1962年4505年Fi（4）pFk（m）p Fi（+3）p Fj（\000）p Fi（1）p 1963 4518 241 3 v 20674559（2）2245 4574 y Fv（=）g（）p Fp（000）p Fv（1）25504533 y Fs（2）p Fm（m）p Fs（+1）2770 4574 y Fv（=）h Fp（\000）pFv（1）p Ft（：）p 3653 4842 4 66 v 3657 4780 59 4 v 36574842伏3715 4842 4 66伏162 5037安Fq（Pr）-5 b（o）克34b（of）h（The）-5 b（or）g（em）34 b（1.1.）48 b Fv（=）-17 b Fp（\））pFv（\））45 b（Supp）s（ose）i（即）e Ft（p）g Fv（是）g（的g）h（形式）e（4）pFt（m）31 b Fv（+）f（3）49 b（即）c Ft（p）3473 5001 y Fm（r）35605037 y Fp（jj）k Ft（n）162 5158 y Fv（精确地）33 b（除法）f（带）gFt（r）k Fv（o）s（dd）c（and）h（that）f Ft（n）c Fv（=）g Ft（x）20695122 y Fs（2）2131 5158 y Fv（+）22 b Ft（y）2281 5122 y F（2）23195158 y Fv（.）44 b（出租）31 b Ft（d）d Fv（=）f（gcd）q（\）pFt（x；）17 b（y）t Fv（\），）31 b（w）m（e）j（ha）m（v）m（e）1309 5378y Ft（x）28 b Fv（=）g Ft（dx）1602 5337 y Fl（0）1625 5378 yFt（；）114 b（y）31 b Fv（=）d Ft（dy）2052 5337 y Fl（0）20745378 y Ft（；）114 b（n）28 b Fv（=）g Ft（d）2456 5337 y Fs（2）24955378 y Ft（n）2553 5337 y Fl（0）1918 5706 y Fv（2）p eop%%页码：3 33 2 bop 162 93 a Fv（带）32 b（gcd）p Ft（x）623 56 y Fl（0）64793 y Ft（；）17 b（y）743 56 y Fl（0）765 93 y Fv（\））28 b（=）f（1）33b（和）1571 213 y（）p Ft（x）1664 172 y Fl（0）1687 213 yFv（\））1725 172 y Fs（2）1787 213 y Fv（+）22 b（\）p Ft（y）1975172 y Fl（0）1997 213 y Fv（\）2035 172 y Fs（2）2102 213年y Fv（=）28 b Ft（n）2264 172 y Fl（0）2287 213 y Ft（：）308387 y Fv（因为）c Ft（r）g Fv（is）d（o）s（dd）j Ft（p）k Fp（j）fFt（n）1227 351 y Fl（0）1251 387 y Fv（，）c（so）f（引理）f（1.5）g（隐含）e（即）j（gcd\（）p Ft（x）2654 351 y Fl（0）2678 387 y Ft（；）17b（y）2774 351 y Fl（0）2796 387 y Fv（\））28 b Ft（>）f Fv（1，）d（a）e（con）m（传统）162 641 y（）p Fp（）-17 b Fv（=））32 b（W）-8b（仪式）32 b Ft（n）d Fv（=）e Ft（n）948 605 y Fs（2）948 666y（1）988 641 y Ft（n）1046 656 y Fs（2）1118 641 yFv（其中）34b Ft（n）1458 656 y Fs（2）1531 641 y Fv（has）f（no）f（prime）g（factors）h（of）f（形式）e（4）p Ft（m）23 b Fv（+）f（3.）44 b（It）32 b（su \016ces）162761 y（to）g（sho）m（w）i（that）e英尺（n）792 776 y英尺（2）864 761y Fv（is）g（a）h（总和）f（of）g（t）m（w）m（o）i（平方）45 b（同样）32b（注）h（即）822 981 y（）p Ft（x）915 940 y Fs（2）9151006年（1）977 981年Fv（+）22年Ft（y）1127 940年Fs（2）11231006年（1）1166 981年Fv（\）\（）p Ft（x）1297 940年Fs（2）12971006年（2）1358 981年Fv（+）g Ft（y）1508 940年Fs（2）15041006 y（2）1547 981 y Fv（\））28 b（=）f（\）p Ft（x）1809 996年y F（1）1849 981 y F（x）1904 996 y F（2）1966 981 y Fv（+）22b（y）2112 996 y F（1）2151 981 y F（y）2199 996 y Fs（2）2238981 y Fv（\））2276 940 y Fs（2）2337 981 y F v（+）h（\）p F t（x）2529996年Fs（1）2568 981年Ft（y）2616 996年F（2）2677 981年Fp（\000）g Ft（x）2832 996 y Fs（2）2872 981 y Ft（y）2920 996y F（1）2959 981 y F v（\）2997 940 y F（2）3036 981 y Ft（；）1621201 y Fv（so）45 b（a）f（pro）s（导管）g（of）g（t）m（w）m（o）i（n）m（um）m（b）s（ers）f（即）f（是）h（和）f（的）g（t）m（w）m（o）i（平方）g（是）e（也是）f（a）h（总和）hf（t）m（w）m（o）162 1322 y（平方）500 1286 y Fs（1）5831322 y Fv（Also）29 b（the）h（prime）d（2）i（is）f（a）h（sum）g（of）f（t）m（w）m（o）i（正方形）44 b（It）29 b（th）m（us）h（su\016ces）h（to）d（sho）m（w）i（that）f（if）fFt（p）162 1442 y Fv（is）k（a）g（质数）g（of）g（形式）h（形式）e（4）pFt（m）22 b Fv（+）g（1）33 b（然后）g Ft（p）f Fv（is）g（a）h（总和）f（of）g（t）m（w）m（o） i（平方）308 1562 y（自）1334 1683 y（（）p Fp（\000）pFv（1）1546 1611 y Fk（p）p Fj（\000）p Fi（1）p 1546 1627111 3 v 1586 1668 a（2）1698 1683 y Fv（=）28 b（\（）p Fp（\ 000）pFv（1））2014年1614年Fi（4）p Fk（m）p Fi（+1）p Fj（\000）p Fi（1）p2014年1627 241 3 v 2119 1668 a（2）2296 1683 y Fv（=）g（+1）pFt（；）162 1857 y Fp（\000）p Fv（1）40 b（is）g（a）g（平方）i（mo）s（dulo）cFt（p）p Fv（；）44b（即）e（存在）f（存在）g Ft（r）i Fv（例如）m（h）f（即）Ft（r）2624 1821 y Fs（2）2704 1857 y Fp（\021）h（\000）p Fv（1）j（\（mo）s（d）32b Ft（p）p Fv（\）。）67 b（T）-8 b（aking）162 1978年y Ft（n）28b Fv（=）f Fp（b）395 1915 y（p）p 479 1915 49 4 v 479 1978a Ft（p）p Fp（c）h Fv（in）g（引理）e（1.3）i（w）m（e）i（见）f（that）f（there）h（are）f（in）m（tegers）h Ft（a）；）17 b（b）29 b Fv（suc）m（h）g（即f（0）gFt（<）f（b）i（<）3406 1915 y Fp（p）p 3489 1915 V 63 x Ft（p）fFv（and）1315 2119 y Fn（\014）1315 2179 y（\013）1315 2239y（\014）1315 2298 y（\014）1349 2264 y Fp（\000）1437 2196y Ft（r）p 1436 2241 V 1436 2332 a（p）1517 2264 y Fp（\000）16272196 y Ft（a）p 1627 2241 52 4 v 1632 2332 a（b）1688 2119y Fn（\014）1688 2179 y（\014）1688 2239 y（\014）1688 2298y（\014）1749 2264 y Fp（\024）2012 2196 y Fv（1）p 1864 2241345 4 v 1864 2332 a Ft（b）p Fv（（）p Ft（n）23 b Fv（+）f（1））22462264 y Ft（<）2422 2196 y Fv（1）p 2360 2241 174 4 v 23602332 a Ft（b）2401 2270 y Fp（p）p 2484 2270 49 v 62 xFt（p）2543 2264 y（：）162 2543 y Fv（If）32 b（w）m（e）i（写入）17022663 y Ft（c）28 b Fv（=）f Ft（r）s（b）22 b Fv2837年Fv（当时）1459 2982年Fp（j）p Ft（c）p Fp（j）27 b Ft（<）17392915 y（pb）p 1697 2959 174 4 v 1697 3050 a（b）1738 2988y Fp（p）p 1822 2988 49 4 v 1822 3050 a Ft（p）1908 2982y Fv（=）2063 2915 y Ft（p）p 2022 2959 132 4 v 2022 2988a Fp（p）p 2105 2988 49 4 v 62 x Ft（p）2191 2982 y Fv（=）22952915 y Fp（p）p 2378 2915 V 67 x Ft（p）162 3216 y Fv（and）15833336 y（0）h Ft（<）f（b）1804 3295 y Fs（2）1866 3336 y Fv（+）cFt（c）2007 3295 y Fs（2）2074 3336 y Ft（<）k Fv（2）p Ft（p:）1623510 y Fv（But）33 b Ft（c）27 b Fp（\021）h Ft（r）s（b）45 bFv（（mo）s（d）32 b Ft（p）p Fv（\），）g（so）952 3730 y Ft（b）9933689 y Fs（2）1055 3730 y Fv（+）22 b Ft（c）1195 3689 y F（2）12623730年Fp（\021）28 b Ft（b）1408 3689年Fs（2）1470 3730年Fv（+）22 b Ft（r）1615 3689 y Fs（2）1655 3730 y Ft（b）16963689年Fs（2）1763 3730年Fp（\021）28 b Ft（b）1909年3689年Fs（2）1949 3730 y Fv（1）22 b（+）g Ft（r）2203 3689 y Fs（1）22423730 y Fv（\））28 b Fp（\ 021）g Fv（0）99 b（\（mo）s（d）32 b Ft（p）pFv（\）p Ft（：）162 3950 y Fv（Th）m（us）i Ft（b）450 3914 yFs（2）512 3950 y Fv（+）22 b Ft（c）652 3914 y Fs（2）719 3950y Fv（=）28 b Ft（p）p Fv（.）p 3653 3950 4 66 v 3657 388859 4伏3657 3950伏3715 3950 4 66伏162 4239 a Fx（1.2）135b（计算）46 b Fh（x）e Fx（和）h Fh（y）162 4424 y Fv（Supp）s（ose）iFt（p）g Fv（is）f（a）g（质数）f（of）h（形式）f（4）p Ft（m）31b Fv（+）h（1.）85 b（There）48 b（is）e（a）g（construction）g（of）g（勒让德）1624544 y（共）34 b Ft（x）g Fv（和）g Ft（y）j Fv（即）d（是）g（解释）g（on）g（第页）h（120｛121）d（of）i（Da）m（v）m（enp）s（ort.）48 b（Im）34b（uncon）m（vinced）i（that）1624665 y（it）41 b（is）g（an）m（y）i（more）f（e \016cien）m（t）g（than）g（the）h（follo）m（wing）c（naiv）m（e）j（算法：）60b（计算）3200 4578 y Fn（p）p 3299 4578 266 4 v 3299 4665a Ft（p）23 b Fp（\000）f Ft（x）3525 4636 y Fs（2）3607 4665y Fv（for）162 4785 y Ft（x）28 b Fv（=）g（1）p Ft（；）17 b FvFt（；）g（：）g（（：）g（：）c（：）27 b Fv（un）m（til）h（it's）h（an）g（in）m（teger.）42b（这个）29 b（tak）m（es）h（at）f（most）2372 4723 y Fp（p）p 24554723 49 4 v 62 x Ft（p）g Fv（步骤）43 b（这里）31 b（a）e（简单）e（p）-8b（ARI）162 4905 y（程序）31 b（whic）m（h）i（implemen）m（ts）e（this）i（算法）p 162 4973 1425 4 v 274 5035 a Fg（1）311 5065y Ff（This）e（代数）e（iden）n（tit）n（y）h（is）h（秘密）e（the）i（断言）e（即）i（标准）g（映射）g Fe（N）37 b Ff（：）28b Fd（Q）5 b Ff（（）Q Fe（i）p Ff（））3135 5035 y Fc（\003）32075065 y Fb（！）28 b Fd（Q）3376 5035 y Fc（\003）3451 5065 yFf（发送）162 5164 y Fe（x）19 b Ff（+）f Fe（iy）30 b FfFe（x）19 b Ff（+）f Fe（iy）s Ff（\）\Ff（\））23 b（=）f铁（x）1241 5134 y Fg（2）1298 5164 y Ff（+）cFe（y）1425 5134 y Fg（2）1489 5164 y Ff（is）28 b（a）f（同态）19185706年Fv（3）p eop%%页码：4 44 3 bop 162 93 a Fa（{两平方和\（n））56 b（=）316213年（当地）316 333 y（适用于\（x=1，楼层\（sqrt \（n）\），）469454年（y=平方米（n-x^2）；）469 574 y（如果（y地板\（y）==0，）g（返回\（[x，地板\（y\）]\））316 694 y（\）；）316 815 y（错误\（n，“）d（is）f（不是）g（a）g（总和）g（共）g（两）g（正方形）162 935 y（}）1621268 y Fu（2）161 b（总和）52 b（总数）i（更多）f（平方）162 1487y Fv（Ev）m（ery）31 b（天然）d（n）m（um）m（b）s（er）h（is）f（a）h（总和）g（of）fFr（四）i Fv（正方形）44 b（见）29 b（第页）h（124｛126）d（of）i（Da）m（v）m（enp）s（ort）g（for）162 1607 y（a）j（pro）s（of）308 1728 y（a）h（天然）f（n）m（um）m（b）s（er）h（is）g（a）g（总和）f（of）h Fr（三）g Fv（平方）h（if）e（和）h（仅）g（如果）e（it）h（is）h（不是）g（a）f（p）s（o）m（w）m（er）i（of）f（4）1621848年（时间）38 b（a）h（n）m（um）m（b）s（er）h（即）f（是）g（同余）m（t）h（到）f（7） g（mo）s（dulo）f（8.）63 b（f）-8 b（或）39 b（例如）i（7）e（is）f（not）i（a）f（总和）g（of）1621969 y（三）33 b（平方）45 b（本）33b（is）f（more）g（di）g（to）g（pro）m（v）m（e.）1918 5706y（4）p人%%拖车结束userdict/end-hook已知{end-hook}如果%%电动势