æãã®ç²Ÿç¥ãæŽãã
ãããŠãçµéšãå°é£ãªéã¡ã®æ¯åã
ãããŠå€©æããã©ããã¯ã¹ã®åã
ãããŠãå¶ç¶ãç¥ã®çºæè ...
A.S. ããŒã·ãã³
åå ãã代ããã«
mod 6ããã³mod 4ã®äœ¿çšã«åºã¥ãæ°å€ã®å æ°å解ã®åé¡ã®è§£æ±ºçãæ瀺ããããšæããŸããããã«ããã2次äŸåæ§ã®ç·åœ¢äŸåæ§ãžã®å€æãæ¯é ããæ³åãèŠã€ããããšãã§ããŸããã
çºèŠãããèŠåæ§ã«åºã¥ããŠãèè ã«ãããšãããã°ã©ã ãæžããšãã«ãæ°åŠã®é確çç決å®æ³ã䜿çšããŠæ°å€ãå æ°å解ããéã®æéã³ã¹ããå€§å¹ ã«åæžããå¯èœæ§ãéãææ³ãæžãããŸããã
ãã®ãã¯ããã¯ã«ããã°ãããã°ã©ã ã¯ããã°ã©ããŒã«ãã£ãŠæžãããŸãã-ç¬åŠã®Belykh Sergey Alekseevichãããã®æå¹æ§ã瀺ããŸããã æ®å¿µãªãããããã¯å€§ããªæ°ã«é©å¿ããŠããŸããã ãã®ææ³ã¯ãããã°ã©ã ãã³ã³ãã€ã«ããããã®ã¢ã«ãŽãªãºã ãšããŠèª¬æä»ãã§æžãããŠããŸãã ã¢ã«ãŽãªãºã ã¯è¡šåœ¢åŒã§æ瀺ãããŸãã
ã¢ãŒã6ã®æ°å€ã®åº§æšãš
ã¢ãŒã4ã§ã¯ããããã®ã¢ãžã¥ãŒã«ã«ããæ°å€éã®çžé¢äŸåæ§ãä¿èšŒãããŸãã N4ã
ã¢ãŒã6ã®åº§æšç³»ã䜿çšããå Žåãšã¢ãŒã4ã®åº§æšç³»ã䜿çšããå Žåã®äž¡æ¹ã§ããã¹ãŠã®åææ°ã4ã€ã®è±¡éã«ããããšãæãåºããŠãã ããã
åããŒãã«ã¯ã16ã®å¯èœãªãªãã·ã§ã³ã®ãã¡4ã€ã«å¯ŸããŠã³ã³ãã€ã«ãããŸãã ãªã16ã®ãã¡ïŒ
å象éã«ã¯ååç©çªå·ãå«ãŸããŠããã4ã€ã®èšèšãªãã·ã§ã³ãç¹åŸŽã§ãã
èšèšãªãã·ã§ã³ã¯ã座æšã®ããªãã£ãããã³ç°ãªãããªãã£ã®åº§æšå€ã®æ¯çã«äŸåããŸãã
æé ã®è©³çŽ°ã«ã€ããŠã¯ãcatã«åŸã£ãŠãã ããã
æåã®æ°åã®æ°ãå«ãè¡šã«ç®ãåããŸãããã®æ°ã®åœ¢åŒã¯æ¬¡ã®ãšããã§ããN= 6 * xy + x + yïŒ
è¡š1ïŒAïŒ
y \ x | 1 | 2 | 3 | 4 |
---|---|---|---|---|
1 | 8 | 15 | 22 | 29æ¥ |
2 | 15 | 22 | 41 | 54 |
3 | 22 | 41 | 54 | 79 |
4 | 29æ¥ | 54 | 79 | 104 |
ãããã£ãŠãããŒãã«ã®ä»»æã®ã»ã«ã«å ¥åã§ããŸãã ãããã£ãŠãçªå·N 1ãè¡š1ïŒAïŒã«ããçªå·L 1ã¯ã次ã®ããã«è¡šãããšãã§ããŸãã
L 1 = 6ïŒ6xy + x + yïŒ+ 1
ããã§ãN 1 = 6 xy +ïŒx + yïŒã
åæã«ããã®è¡šã®æ°å€ã®åº§æšã¯äž¡æ¹ãšãæ£ã§ãããå¶æ°ã§ãå¥æ°ã§ãããŸããŸããã 座æšã®åäžæ§ãããã³ãããã®åã®ç¬Šå·ã¯ãmod 6ã§ã³ã³ãã€ã«ããã座æšç³»ã®åº§æšãšmod 4ã§ã³ã³ãã€ã«ããã座æšç³»ã®åº§æšãšã®éã®çžé¢äŸåé¢ä¿ãèšç®ããã¢ã«ãŽãªãºã ã«åœ±é¿ãããããããã«æ³šæãåèµ·ããŸãããããã®ã¢ãžã¥ãŒã«ã«ãã£ãŠèšç®ãããæ°å€éã®äŸåé¢ä¿ã ããã¯ãèæ ®ããããªãã·ã§ã³ã®å æ°å解ã«éããããããäž»ãªå åã§ãã
ãã®ãããããŒãã«ã®è¡çªå·ã¯ã笊å·ãåé¡ã®ããŒãã«ã®çªå·ãšåçªå·ã«äŸåãã座æšã§ãã è¡å ã®é£æ¥ããæ°åã®æ°ã®å·®ã¯äžå®ã§ãã è¡ã®å¢åã®å·®ãåæ§ã§ãã mod 6ã§ã³ã³ãã€ã«ãããããŒãã«ã®å Žåããã®å€ã¯6ã§ããmod4ã§ã³ã³ãã€ã«ãããããŒãã«ã®å Žåããã®å€ã¯4ã§ãã
ããŒãã«1ïŒAïŒãã³ã³ãã€ã«ãããšãã«ãèšç®ã®ããã®æåã®æ°éã®å€ïŒ1,2ã3,4ã5,6 ...ããã®åŸãæåã®è£å©çªå·ã·ãªãŒãºã®çªå·ã®ããŒãã«ãã³ã³ãã€ã«ããå Žåããã®æ°ã¯æ¬¡ã®ããã«è¡šãããŸãïŒN sub> 3 = 6xy- xyïŒ-1ã-2ã-3ã-4ã-5ã-6 ...
è¡š3ïŒCïŒ
y \ x | 1 | 2 | 3 | 4 |
---|---|---|---|---|
-1 | 4 | 9 | 14 | 19 |
-2 | 9 | 20 | 31 | 42 |
-3 | 14 | 31 | 48 | 65 |
-4 | 19 | 42 | 65 | 88 |
2çªç®ã®è£å©ç³»åã®æ°ã«ã€ããŠã¯ã次ãååŸããŸãã
è¡š2ïŒBïŒ
y \ x | 1 | 2 | 3 | 4 |
---|---|---|---|---|
1 | 6 | 11 | 16 | 21 |
2 | 13 | 24 | 35 | 46 |
3 | 20 | 37 | 54 | 71 |
4 | 27 | 50 | 73 | 96 |
è¡š4ïŒDïŒ
y \ x | 1 | 2 | 3 | 4 |
---|---|---|---|---|
-1 | 6 | 13 | 20 | 27 |
-2 | 11 | 24 | 37 | 50 |
-3 | 16 | 35 | 54 | 73 |
-4 | 21 | 46 | 71 | 96 |
åæ§ã«ãmod 4ã®2çªç®ã®æ°å€è£å©ã·ãªãŒãºã®ããŒãã«ãèšç®ãããŸãã
ç¹å®ã®äŸãæ€èšããŸãã
䜿çšãããã¢ãžã¥ãŒã«ã«ãã£ãŠèšç®ãããæ°å€ã®æ°ã¯ã䜿çšããã座æšç³»ãšåã象éã«å±ããŠããŠããç°ãªã象éã«å±ããŠããŠãããŸããŸããã ãããåæã«ããããã¯åžžã«ãããã®éã®å³å¯ãªçžé¢äŸåé¢ä¿ã«ãããŸãã
ãŸããå³å¯ãªçžé¢äŸåæ§ã«ã¯ããããã®ã¢ãžã¥ãŒã«ã«åŸã£ãŠèšç®ããã座æšããããã®åèšãããã³ãããã®å·®ã®ç©ããããŸãã
L = 10525ãªã©ã®æ¹æ³è«ã®èŠåæ§ãèæ ®ããŠãã ããã ãã®çªå·ãå±ããè£å©çªå·ã·ãªãŒãºã決å®ããŸãã æ°å€ãæåãŸãã¯2çªç®ã®æ°å€è£å©çŽæ°ã«å±ãããšããæ¡ä»¶ã¯ãmod 6ã§ãã®æ°å€ã®ïŒ+1ïŒå°äœã¯ã©ã¹ã«å±ããããšããŸãã¯ïŒ-1ïŒmod 6ã§å°äœã¯ã©ã¹ã«å±ããããšã§ãã
N 6 =ïŒ10525-1ïŒ/ 6 = 1754;
çªå·ã¯ãçªå·ã®æåã®è£å©ã¯ã©ã¹ãæããŸããã€ãŸããæåã®ïŒAïŒãŸãã¯3çªç®ïŒCïŒã®ããŒãã«ã«å±ããããšãã§ããŸãã
次ã®ã¹ãããã¯ãæ€èšããããšã§ãïŒèæ ®ãããæ°ã®åº§æšã¯ã©ã®ããªãã£ãæã€ããšãã§ããŸããïŒ æ°å€ã¯å¶æ°ãªã®ã§ããã®ããŒãžã§ã³ã§ã¯äž¡æ¹ã®åº§æšãåãããªãã£ãæã€ããšãã§ããŸãã äž¡æ¹ã®åº§æšãå¶æ°ã§ãããšä»®å®ããŸãã ãã®å®æœåœ¢æ ã§ã¯ãïœïœïœ ïŒã«åŸã£ãŠèšç®ãããéïŒïœ ïŒ ïœïŒïŒè£æ£å€ïŒããïœïœïœ ïŒã«åŸã£ãŠèšç®ãããè£æ£å€ã®å€ãžã®å€æä¿æ°ã¯ïŒïŒïŒïŒïœ ïŒ ïŒã§ããã
ãã®ä¿æ°ã¯ãmod 6ããã³mod 4ã§èšç®ãããè£æ£å€ã®æ°å€ã·ãªãŒãºãæ¯èŒããããã«èšæ¶ãããŸãã
mod 6ã®æ°å€ã«åºã¥ããŠãmod 6ã®äžé£ã®è£æ£å€ã6ã®ééã§èšç®ãããŸãã æ°åã®æåã®å€ã¯å°äœã¯ã©ã¹ã§ãããmod 6ã§èæ ®ãããæ°ã®æ°ãå±ããŸãã
1754ïŒ6 = 292 * 6 + 2ã
åãåã£ãæ®é«ã«åºã¥ããŠã笊å·ãèæ ®ããŠã6ã®ééã§mod 6ã«äžé£ã®æ°å€ã®è£æ£å€ãäœæããŸãã
2 8 14 20 26 32 38 44 ⊠(1)
次ã«ãmod 4ã§æ°å€ã決å®ããŸãïŒmod 6ã§æ°å€ã決å®ããã®ãšåæ§ïŒã
N 4 =ïŒ10525-1ïŒ/ 4 = 2631;
mod 4ã®æ°å€ã«åºã¥ããŠãéé4ã§mod 4ã®è£æ£ââå€ã®æ°å€ã·ãªãŒãºãèšç®ããŸãã
æ°åã®æåã®å€ã¯å°äœã¯ã©ã¹ã§ãããèæ ®ãããæ°ã®çªå·ã¯mod 4ã«å±ããŸãïŒå¶æ°åº§æšã®åèšã§è¡šãããè£æ£å€ãå€æããå Žåãå€æã®çµæãšããŠåŸãããè£æ£å€ã¯ãè£æ£å€ã®åã®ç¬Šå·ãä¿æããŸãïŒã
2631ïŒ4 = 657 * 4 + 3ã
åãåã£ããã©ã³ã¹ã«åºã¥ããŠã笊å·ãèæ ®ããŠã4ã®ééã§mod 4ã«äžé£ã®æ°å€ã®è£æ£å€ãäœæããŸãã
3 7 11 15 19 23 27 31 35 39âŠ
çžé¢ä¿æ°1 / k 6ã«åºã¥ããŠãmod 4ã«åŸã£ãŠèšç®ãããè£æ£å€ã®æ°å€ã·ãªãŒãºã®å€ãmod 6ã«ããè£æ£å€ã«å€æããŸãã
2 4,666 7,333 10 12,666 15,333 18 20,666 26 ⊠(2)
æ°å€ã·ãªãŒãºïŒ1ïŒãšïŒ2ïŒã®æ¯èŒã«åºã¥ããŠãmod 6ã«åŸã£ãŠ24ã®ééã§èšç®ãããè£æ£å€ã®æ°å€ã·ãªãŒãºãæ§ç¯ããŸãã
2 26 50 74 98 âŠ
ããã§ãã¢ãžã¥ãã§èšç®ãããè£æ£å€ã®æ°å€ã·ãªãŒãºã«åºã¥ããå€å¥åŒã®æ°å€ã·ãªãŒãºãæ§ç¯ããããã«å¿ èŠãªãã¹ãŠã®ããŒã¿ãæ¢ã«24ã«ãªããŸããããããŠã察å¿ããå€å¥åŒã¯ãå¿ ç¶çã«ãèæ ®ãããæ°ã®æŽæ°åº§æšã®æ±ºå®ã確å®ã«ããŸãã
å®éãæ°å€ãšç¹å®ã®ä¿®æ£å€ã«åºã¥ããŠã座æšã®æšå®ç©ããããã«VietaåŒã«åŸã£ãŠæ±ºå®ã§ããŸãã
D =ïŒx + yïŒ^ 2 /ïŒ2 ^ 2ïŒ-4 * xy;
ãã®çµæã以äžãåŸãããŸãã
-291 -119 341 1089 2125 3449 5061 6961
ãã¿ãŒã³ã®åæã«ãããè¡š10-1ã®ããŒã¿ã®èæ ®äºé ã䜿çšããèæ ®äºé ã®çµæãåŸãããŸããã
åãšè¡ã®äž¡æ¹ãããã³ã»ã«ã®çµæãèæ ®ããŠãè¡šããŒã¿ãèæ ®ããŸãã
1åç®ã2åç®ã3åç®ã«ã¯ãç¹å®ã®å€å¥åŒã«åŸã£ãŠèšç®ã«äœ¿çšããããå€æŽããã段éçãªä¿®æ£å€ïŒa i ïŒãå«ãŸããŠããŸãã1åç®ã2åç®ã3åç®ã
æåã®åã®è£æ£ã¯ã-4ã«çããéã ãæåã®è£æ£å€ããéå§ããŠã段éçã«å®è¡ãããŸãã
2åç®ã§ã¯ã調æŽã¹ãããã®å€ã24å¢å ããŸãã3åç®ãåæ§ã§ãã 4åç®ã5åç®ã6åç®ã¯æ¬¡ã®åŒã§èšç®ãããŸãã
D i -[ïŒa i ïŒ/ 2] 2 ïŒB i ïŒ
åé¡ã®åè¡ã«å¯ŸããŠã
7åç®ã¯ãä¿®æ£å€ã®èª¿æŽãç¹å®ã®è¡ã®é£æ¥å€ãèæ ®ãããèšç®ããã2ã€ã®éã®å·®ïŒPïŒã§ãã
8çªç®ã®åã¯ãç¹å®ã®è¡ã§ãæåã«èšç®ãããå€å¥åŒã®ä¿®æ£å€ãå·®ïŒPïŒã§å²ã£ãåã§ãã
ããã«ãèšç®ã§åŸãããæŽæ°ã®åã«ããã座æšã®åèšã«çããè£æ£å€ãšèæ ®ãããæ°ã«å¯Ÿå¿ããyã®å€ã®äž¡æ¹ã決å®ã§ââããŸãã äžããããäŸã§ã¯ã2 -3 * 24 =-y; 2-ïŒ-3 * 24ïŒ= 74 =ïŒx + yïŒã
ãã¹ãŠã®è³ªåã«çãããšã¯æããªãã§ãã ããã çŸåšã®æ³åŸãçãã§ããããšã«åæããŸãã è¿œå ã®èšç®ãªãã§ã¯ãçã®çãã¯å°é£ã§ãã èšç®ã«ã¯ã質åã«çããããã«å€æŽãå ããããšãã§ããããã°ã©ã ãå¿ èŠã§ã
è¡š10-1
y \ x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
1 | 2 | 26 | 50 | -292 | -288 | -284 | 4 | -73 |
2 | 22 | 46 | -292 | -240 | -188 | 52 | 52 | -5.615 |
3 | -6 | 18 | 42 | -300 | -200 | -100 | 100 | 3 |
4 | -10 | 14 | 38 | -316 | -168 | -20 | 148 | -2,135 |
æ¹æ³è«ãžã®æ³šé
mod 6ããã³mod 4ã®èªç¶æ°åããã®è£å©æ°å€ã·ãªãŒãºã®äœ¿çšã«åºã¥ããŠãååç©æ°ã¯16åã®ã°ã«ãŒãïŒããªã¢ã³ãïŒã«åããããåå²ãããŸãã 䞊åèšç®ãæ¯èŒããããšã«ãããèæ ®ãããæ°å€ãè€åæ°å€ã®ææ¡ãããã°ã«ãŒãïŒææ¡ããããªãã·ã§ã³ïŒã«å±ãããã©ãããå€æããããšãã§ããŸããã
æ°å€ãåæããŠåçŽãã©ãããå€æããå Žåãèšç®ã®æ倧æ°ã¯4ã§ãã èšç®ãªãã·ã§ã³ã«ã€ããŠã¯ã以äžã§èª¬æããŸãã æ°åã®ã°ã«ãŒãããšã«ãæ°åã®åæã圢åŒåããèšç®ã¢ã«ãŽãªãºã ãå®åŒåãããŸãã èæ ®ãããæ°ã®èŠå ãèŠã€ããããã«ã2ã€ã®æªç¥æ°ãæã€2次æ¹çšåŒã解ãããã®ä»£æ¿æ¹æ³ãéçºãããŸããã ãã®æ¹æ³è«ã¯ãäºæ¬¡äŸåé¢ä¿ã®ç·åœ¢ãžã®å€æã®èŠã€ãã£ãèŠåæ§ã«åºã¥ããŠãããèè ã«ããã°ãããã¯æéã³ã¹ãã®èšç®ãå€§å¹ ã«åæžããŸãã
äºæ¬¡æ¹çšåŒã解ãå¥ã®æ¹æ³ã¯ãå€å¥åŒãéžæããå¹³æ¹æ ¹ãé£ç¶ããŠæœåºããããšã§è§£ãæ±ããåŸæ¥ã®æ¹æ³ãšã¯å¯Ÿç §çã«ãææ¡ãããæ¹æ³ã§ã¯ãä¿®æ£ãããå€å¥åŒã®æŽæ°éçŽæ§ïŒD i ïŒãšãããã®å·®ïŒD ïŒi + 1ïŒ -D i ïŒãæåã®å€ã2çªç®ã®å€ã§å²ããšãã«æŽæ°ã®åãèŠã€ããŸãã
ãã®ã¢ãããŒãã䜿çšãããšãããªãã®æ°ã®èšç®ã§ã®åæã§ãåæã®éèŠã§ãªãå€ã䜿çšã§ããŸãã èšç®ãªãã·ã§ã³ããšã«ãExcelããŒãã«ã®æ©èœã䜿çšããããã°ã©ã ãã³ã³ãã€ã«ãããŠããŸãã èè ã«ãããšãããã°ã©ã ãæžããåŸã®ææ³ã«ãããæ°è«ã®æ°ãããã¿ãŒã³ãèŠã€ããããšãã§ããããã«ãªããŸããããã¯ãæ°åãå æ°å解ããéã®æéã³ã¹ãã®åæžã«ããã«åœ±é¿ããã¯ãã§ãã
質åã¹ããŒã¿ã¹ïŒæ¹æ³è«ã®çŽ¹ä»ïŒ
å®éã«äžããããæ°ãçŽ å æ°ã«å解ããåé¡ã¯ãæ°åŠã§æãé£ããåé¡ã®1ã€ã§ãã çŽå å3äžçŽ Erastofenã¯ãäžããããå¶éAãããå°ãããã¹ãŠã®çŽ æ°ãèŠã€ããæ¹æ³ãææ¡ããŸããã20äžçŽåé ã«Brunã«ãã£ãŠæ¹åãããåŸããã®æ¹æ³ã¯ãã³ã³ãã¥ãŒãã£ã³ã°ããã€ã¹ã䜿çšããã«ãã®åé¡ã解決ããå¯äžã®æ¹æ³ã§ãã çŽ æ°ã®ååžã®æ³åãèŠã€ããä»ã®è©Šã¿ã¯ãéèŠãªçµæããããããŠããªãã
ææ¡ãããäœæ¥ã¯ãè€åæ°ã®ååžã®æ³åã«åºã¥ããã®åé¡ã®è§£æ±ºçã®å£°æã§ãã
äœæ¥ã¯ãçŽ æ°ã§ã¯ãªãçŽ æ°ã®å·®ã®å åãæ€çŽ¢ãããããã®å åã䜿çšããŠçŽ æ°ãèæ ®ããããã©ãããå€æããããšã«ãããŸãã
ãã®åé¡ã®è§£æ±ºçã¯ã2ã€ã®æªç¥æ°ãå«ã2次æ¹çšåŒã®è§£æ±ºçã«ãã£ãŠåæžãããŸããã éåžžããã®ãããªæ¹çšåŒã¯æªç¥ã®1ã€ïŒãã®å Žåã¯kïŒãéžæããããšã§è§£æ±ºãããŸããããã®æ¹æ³ã¯èšç®ã¹ãããã24ã«å¢ããã«ãããããããå€æ°ã®å Žåã¯ããªãé¢åã§ãã
ã¡ãœããã®æ¬ é¥
ãœãããŠã§ã¢ãšPCã䜿çšããªãé£æ床ãšåŽåã
ãã®æ¹æ³ã®å©ç¹
åé¡ã®æ°ã®æåæ°ã«é¢ä¿ãªããPCãšãœãããŠã§ã¢ã䜿çšããæ©èœã çŸåšã®ç 究ã§ã¯ãç§ãã¡ã®æèŠã§ã¯ãæ°è«ã®åéã§ããŸããŸãªåé¡ã解決ããã®ã«é©ãããé©åãªå®éšå®€ã®åºç€ãäœæãããŸãããããšãã°ãæŽæ°ã®ç«æ¹äœã1次ãš2次ã®é ã®åèšãšããŠè¡šçŸããŸãã ç§ãã¡ã¯ãPCã®ã°ã«ãŒããä»ããŠæ°åã«ããªãã®æ°ã®æ°åãæãããšãæ°åãæ¶ããããšãããããšãç¥ã£ãŠããŸãã éçºããææ³ã§ã¯ãèšç®ã«å°æ°æ¡ã®æ°åã䜿çšããããããPCã°ã«ãŒãã䜿çšããå¿ èŠã¯ãããŸããã å¿ èŠãªå¯äžã®é£ç¶èšç®ã¯ãæ°å€ã®é€ç®ã§ãã
æ€èšäžã®æ¹æ³è«ã«åŸã£ãŠããã°ã©ã ãäœæããéã®åé¡ã«ãããèè ã¯ãæ¹æ³è«ã«åºæã®ååãããççž®ããã圢åŒã§èª¬æããããšããã®ã¯æªããªãã ãããšããæèŠã«è³ããŸããã çµå±ã®ãšããããã®è©Šã¿ã¯ãããããããã°ã©ããŒã ãã§ãªããèŠã€ãã£ãèŠåæ§ã«ç²ŸéããŠããªãæ°åŠè ã«ãèå³ãããã§ãããã
ãããã£ãŠããã®ãããªèª¬æã®è©Šã¿ã¯ãæ¹æ³è«çãªãã·ã§ã³ã®1ã€ãèæ ®ããããšã§äœæ¥å šäœã説æããå¿ èŠããããšããäºå®ãšãããã°ã©ãã«ãšã£ãŠã圹ã«ç«ããªããããããªãèŠåæ§ã«åºã¥ããŠããŸãã äœæ¥ã®æå³ãç解ãããšããæ¹æ³è«ã®ãã¹ãŠã®éšåã¯ç解ã®ããã«æ¯èŒã§ããŸãã
ãã¯ããã¯ãæžãç®ç
ææ¡ãããä»äºã®äž»ãªç®çã¯ãäžããããæ°LãçŽ æ°ã§ãããã©ããããŸãã¯ããã1ã«çãããªãå°ãªããšã2ã€ã®åçŽãªå åã®ç©ã§ãããã©ãããå€æããããšã§ããæå°æªã æåã®æ®µéã§ã¯ãã¢ãžã¥ãæ¯èŒæ³ãé©çšããŸããã
ä»»æã®æ°Lã¯æ¬¡ã®åœ¢åŒã§è¡šãããšãã§ããŸãã
L = m N + rã
ããã§ïŒ
mã¯æå®ãããã¢ãžã¥ãŒã«ã§ãã
N-çªå·ãåŒã³åºããŸããã
rã¯å°äœïŒæ£ãè² ã0ã«çãããªãããšããããŸãïŒãrã¯-ïŒm-1ïŒãïŒm-1ïŒã®ç¯å²ã«ãããŸãïŒm = 6ã-5ã+ 5ïŒã
m = 6ãéžæããŸããããã«ãããåæã®å¯Ÿè±¡ãšãªãç¡éã®æ°ãããå å2ã3ã6ãååšããæ°ãé€å€ããæ©äŒãäžããããŸãã åæã®å¯Ÿè±¡ãšãªããã¹ãŠã®æ°å€ã¯ãèªèãã«ãããšåœç€Ÿã«ãã£ãŠåŒã°ããŠããŸãã
ã¢ãžã¥ãŒã«6ã«ã€ããŠã¯ãæ®ãã«å¿ããŠããã¹ãŠã®èªç¶æ°ã6ã€ã®ã¯ã©ã¹ã«åé ãããŸãã
1ïŒr =0ãL= 6 NïŒã€ãŸãããã®ã¯ã©ã¹ã®ãã¹ãŠã®æ°åã¯ã¢ãžã¥ãŒã«Mèªäœãæ§æããŸãïŒ
2ïŒr = 1; L = 6 N + 1ãããã¯r = -5ãšåçã§ãã L = 6N-5ã
3ïŒr = 2; L = 6 N + 2ãããã¯r = -4ãšåçã§ãã L = 6N-4ã
4ïŒr = 3; L = 6 N + 3ãããã¯r = -3ãšåçã§ãã L = 6N-3ã
5ïŒr = 4; L = 6 N + 4ãããã¯r = -2ãšåçã§ãã L = 6N-2ã
6ïŒr = 5; L = 6 N + 5ãããã¯r = -1ãšåçã§ãã L = 6N-1ã
-*ã¢ãžã¥ãŒã«Mã¯ãæå®ãããæ°å€mã®åæ°ã§ãããã¹ãŠã®æ°å€ã®ã·ã¹ãã ã§ãã æ°mã¯ãäžããããã¢ãžã¥ãŒã«Mã®æå°æ°ã§ããæ°aãšbãmã§å²ã£ããšãã«åãæ®å·®ãåŸãããå Žåããããã¯mãæ³ãšããæ¯èŒå¯èœãªãã®ãšåŒã°ããŸãã ããã¯æ¬¡ã®ããã«æžãããŠããŸãã
aÎbãïŒmod mïŒ
ãã®aãšbã®é¢ä¿ã¯ãmãæ³ãšããæ¯èŒãšåŒã°ããŸãã ä»»æã®æ°ã¯ããã®æ°ãmã§é€ç®ãããšãã®å°äœã§mãæ³ãšããŠæ¯èŒã§ããŸãã äŸïŒaãmã§é€ç®ãããšå°äœ1ãåŸãããå ŽåïŒ
aÎ1; ïŒmod mïŒ
åèšãmã§å²ãåããããã«1ãaã«è¿œå ããå¿ èŠãããå Žåã
Î-1; ïŒmod mïŒ
ãã®å Žåã-1ã¯ãè² ã®å°äœããšåŒã°ããŸãã ç§ãã¡ã«ãšã£ãŠèå³æ·±ãã®ã¯ã2ã€ã®ã¯ã©ã¹ã®æ°ã§ãã
LïŒ+1ïŒ= 6N + 1; LïŒ+1ïŒÎ1ïŒmod 6ïŒ[1]ã
ããããæ¯åºçªå·ïŒ+1ïŒãšåŒã³ãŸããã ãããŠ
LïŒ-1ïŒ= 6N-1; LïŒ-1ïŒÎ-1ïŒmod 6ïŒ[2]ãã
ãã©ã³ãçªå·ïŒ-1ïŒãšåŒã³ãŸããã
ãããã®æ°å€ã¯å¥æ°ã§ããã3ãŸãã¯6ã§å²ãåããŸããã ã€ãŸãããããã¯æ°åãèªèããã®ãé£ããã§ãã é¡æšã«ãããmod 4ã䜿çšããå Žåã«ãæ°å€ãèæ ®ãããŸãã
第1è£å©çªå·ã·ãªãŒãºã®ãã¹ãŠã®é£ããçªå·ã®çªå·ãå«ãããŒãã«ã®ç·šé
èªç¶æ°ç³»åã®ãã¹ãŠã®å°é£ãªæ°ãäœç³»åããããã«ã4ã€ã®ããŒãã«ãã³ã³ãã€ã«ãããŠããŸãã ã³ã³ãã¯ãã«ããããã«ãããŒãã«ã«æ°å€Lèªäœãå ¥åããã®ã§ã¯ãªããæ°å€Nãå ¥åããŸããå¿ èŠã«å¿ããŠãNãç¥ã£ãŠããã°ãåŒ[1]ã[2]ã§Lãèšç®ã§ããŸãããã¿ãŽã©ã¹ã®ããŒãã«ã®ååã«åŸã£ãŠã
ãã¹ãŠã®ããŒãã«ã«å ±éããç¹æ§ãèæ ®ããŠãã ããã åããŒãã«ã®ãŒãè¡ã§åã«çªå·ãä»ããŸãïŒ1ã2ã3ã4ã...åããŒãã«ã®åã§è¡ã«çªå·ãä»ããŸãïŒ1ã2ã3ã4 ...
ãµã³ãã«è¡šïŒ
y \ x | 1 | 2 | 3 | 4 | 〠| |
---|---|---|---|---|---|---|
1 | x | |||||
2 | x | |||||
3 | x | |||||
4 | x | |||||
... | x | |||||
x i | ã |
åããŒãã«ã®ãŒãã®è¡ãšåã座æšè»žãšããŠäœ¿çšããyãšxã瀺ããŸãã ãã®ã¹ããŒãã¡ã³ãã¯ããã¹ãŠã®åææ°ã4ã€ã®ããŒãã«ïŒç¹å®ã®åº§æšç³»ã®è±¡éïŒã«ãããšããäºå®ã«åºã¥ããŠããŸãã 座æšè»žäžã«ããè¡çªå·ãšåçªå·ã¯ã察å¿ãã笊å·ãæã€èæ ®ãããæ°å€ã·ãªãŒãºã®æ°å€ã®åº§æšã§ãïŒæ°å€Nã®åº§æšã¯æ°å€Lã®åº§æšã§ããããŸãïŒã
ããŒãã«ã®ä»»æã®ã»ã«ã«ãæ°å€L iã®æ°å€N iãå ¥åããŸãã æ°N i ïŒããã³æ°L i ïŒã®åº§æšã¯x i ãy iã§ãã æ°å€L iã¯X iãšY iã®ç©ã§ãã ãããæ£åŒãªåœ¢åŒã§èšè¿°ããŸãã
L i = X i Y i ãããã§X i = 6 x i ±1; i = 6 at i ±1 [3]
L i = 6 N i ±1
N i iã¯è¡šã«ãªã¹ããããŠããŸãïŒãµã³ãã«è¡šãåç §ïŒã
*ããŒãã«ã§x = yã®ã»ã«ãéžæãããŠããŸãã ãããã®ã»ã«ã¯ãåããŒãã«ã®äž»ãªäžé察è§ç·ãæ§æããŸãã ããã¯x = y = 1ããå§ãŸããç¡æéã«ç¶ããŸãã