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±q¯d¨¥ªOºô¤Í Patrick Lau ªº¯d¨¥¤¤¤ÞµoÆF·P¡A¸Õ¬Ý²¦¤ó©w²z (Pythagorean Theorem) ¤¤ªº¯À¼Æ¡C

§Ú­Ì¹ï²¦¤ó©w²z¬Û«H¤£·|·P¨ì­¯¥Í¡A¤]³\¥¦¬O§Ú­Ì©Ò¾Çªº²Ä¤@¹D©w²z¡C³o²¦¤ó©w²z¦b¤¤°ê¤S¦W°Ó°ª©w²z (Soon Go Theorem) ©Î¤ÄªÑ©w²z¡C

¬Û¶Ç¥j§Æþ¼Æ¾Ç®a²¦¹F­ô©Ô´µ (Pythagoras ¬ù«e 569 - ¬ù«e 475) ±qª½¨¤¤T¨¤§Îªº¦aªO¤¤¤ÞµoÆF·P¡Aµo²{ª½¨¤¤T¨¤§Îªº±×Ã䪺¥­¤è¥¿¦n¬O¥t¥~¨âÃ䪺¥­¤è©M¡A§Y¡G

³oÅã²L¦Ó¹ê¥Î©w²z¡A§Ú­Ì¦bªì¤¤¼Æ¾Ç½Ò©w¥²±µÄ²±o¨ì¡C­Y§Ú­Ì¶i¤@¨B³W©w¦¡¤¤ªº a¡Bb¡Bc §¡¬°¥¿¾ã¼Æ (Positive Integer) ¡A³o«K¬O©Ò¿×ªº²¦¹F­ô©Ô´µ¼Æ (Pythagoras Numbers) ©Î²¦¹F­ô©Ô´µ¤T­«¼Æ (Pythagorean Triple)¡A²ºÙ²¦¤ó¼Æ©Î¤ÄªÑ¼Æ¡C

³o¤T¼Æ²Õ¤¤·|§_¥þ¬°¯À¼Æ¡Aµª®×¬O¤£·|¡C

­ì¨Ó¥j§Æþªº¼Æ¾Ç®a¤wª¾­n¨Ï¤T¾ã¼Æ a¡Bb¡Bc ²Å¦X²¦¤ó©w²z¡A¨º¥²¶·¦³¨â¼Æ u ©M v ¨Ï¡G

( a, b, c ) = (u2 - v2, 2uv, u2 + v2)

­Y­n¹F¦Ü¥»­ì¸Ñ (Primitive Solution) §Y GCD ( a, b, c ) = 1¡A§Ú­Ì¥²¶·­n¨D u > v ¡BGCD(u, v) = 1¤Î u¡Bv ¬°¤@©_¤@°¸¡C

¨º»ò 2uv ¥²¬°°¸¼Æ¥B¤£µ¥©ó 2 ¡A·íµM§ä¤£¨ì¥þ¯À¼Æªº¸Ñ¤F¡C

°h¤@¨B¡A¨º u2 - v2 ¥H¤Î u2 + v2 ³£¬O¯À¼Æ¡A¤S¥i¥H¶Ü¡H

§Ú­Ìª¾¹D u2 - v2 = (u - v) (u + v)¡A­n¨Ï u2 - v2 ¬°¯À¼Æ¥u¦³¨Ï u - v = 1 ¤Î u + v ¬°¯À¼Æ¡C

¦p¨ú u = 3 ¤Î v = 2¡A§Ú­Ì¦³ (5,12,13) ³o¤@²Õ¦X¡A²Õ¦X¤¤ªº 5 ©M 13 §¡¬°¯À¼Æ¡C

¤S¦p¨ú u = 6 ¤Î v = 5¡A§Ú­Ì¦³ (11, 60, 61) ³o¤@²Õ¦X¡A¤S§ä¨ì 11 ©M 61 ¨â¯À¼Æ¤F¡C

·|¤£·|¦³¨Ò¥~¡H¯dµ¹¤j®a·Q¤@·Q§a¡A­p¤@­p§a¡C

 

¦P¾lªº´¼¼z

·íµM¤£·|³o¼Ë²³æ§a¡C­Y¨ú u = 9 ¤Î v = 8¡A§Ú­Ì¦³ u2 - v2 = 17¡A³o¬O¤@¯À¼Æ¡A¦ý u2 + v2 = 145¡A¤£¬O¯À¼Æ¡C

¨ä¹ê§Ú­Ì¤F¸Ñ u - v = 1 ¤Î u + v ¬°¯À¼Æ¥u¬O¨ä¤¤¤@±ø¥ó¡A­Y­n¨Ï u2 + v2 ¬°¯À¼Æ¡A§Ú­ÌÁÙ±o¤@¨Ç­n¨D¡A³o¬O¤°»ò¡A¥Î¤@¤BÂI¦P¾lªºª¾ÃÑ«K¥iª¾¡C

³] v = u - 1 ¡A u2 + v2 = u2 + (u-1)2 = 2u2 - 2u + 1

­Y­n¨Ï u2 + v2 ¬°¤@¯À¼Æ¡A§Y­n¨Ï u2 + v2 ¤£µ¥©ó¥ô¦ó¯À¼Æªº­¿¼Æ¡C

§Y¦b¥ô¦ó¯À¼Æ¼Ò p ¤U¡A 2u2 - 2u + 1 §¡¤£µ¥©ó¹s¡C

¬G§Ú­Ì¦Ò¼{¨Ï 2u2 - 2u + 1 = 0 (mod p) ªº¼Æ­È¡A§â¤§¬D°_¡C

2u2 - 2u + 1 = 0 (mod p) => 2u(u-1) = -1 (mod p)

§Ú­Ì¥u»Ý¦Ò¼{ f(u) = 2u(u-1) ¦b¤°»ò±¡ªp·|¦³ f(u) = -1 (mod p)¡C

 

¨ä¹ê p ¤£¥i¬° 2 ¡A¦]¦b¼Ò 2 ¥H¤U 2u(u-1) ùÚ¬°¹s¡C

¦Ó u ¤£¥i¬° 0 ©Î 1¡A¦]³o¤]ùÚ¬°¹s¡C

¦A¥[¤W f(u) ©M f(p+1-u) ªº¼Æ­È¬O¬Û¦P¡A¦¹µ¥³£´î¤Ö¤F§Ú­Ì´ú¸Õ¤§¼Æ¥Ø¡C

 

¤Uªíµ¹¥X¤@¨Ç p ©M u ªº¼Æ­È¨Ï f(u) = -1 (mod p) ¤Î¬ÛÃöªº²¦¤ó¼Æ¤Î¨ä¤À¸Ñ¦¡¡C

p
u (mod p)
u ªº¨Ò¤l
u2 - v2
u2 + v2
u2 + v2 ªº¯À¦]¤l¤À¸Ñ¦¡
5
2
7
13
85
5*17
4
9
17
145
5*29
13
3
16
31
481
13*37
11
115
229
26221
13*2017
17
7
75
149
11101
17*653
11
45
89
3961

17*233

·íµM¨Ò¤l¤£¦¹¤W­z¤T²Õ¡AÁÙ¦³«Ü¦h¡C

¶¶±a¤@´£¡A¦]¥O u - v = 1 ©Ò¥H§Ú­Ì¦³

u2 + v2 - 2uv = (u - v)2 = 1

§Y u2 + v2 ¤ñ 2uv ¦h 1 ¦Ó¤w¡A³oºØ¤T¼Æ¤¤¦³¨â¼Æ¬Û®t¤@ªº±¡ªp¡A§Ú­ÌºÙ§@Åp¥Í²¦¤ó¼Æ (Twin Pythagoras Numbers)¡C

 

 

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