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±q¯d¨¥ªOºô¤Í Patrick Lau ªº¯d¨¥¤¤¤ÞµoÆF·P¡A¸Õ¬Ý²¦¤ó©w²z (Pythagorean Theorem) ¤¤ªº¯À¼Æ¡C
§Ú̹ﲦ¤ó©w²z¬Û«H¤£·|·P¨ì¯¥Í¡A¤]³\¥¦¬O§Ú̩ҾǪº²Ä¤@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§Ú̦bªì¤¤¼Æ¾Ç½Ò©w¥²±µÄ²±o¨ì¡CY§Ú̶i¤@¨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
§Ú̪¾¹D 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§Ú̦³ (5,12,13) ³o¤@²Õ¦X¡A²Õ¦X¤¤ªº 5 ©M 13 §¡¬°¯À¼Æ¡C
¤S¦p¨ú u = 6 ¤Î v = 5¡A§Ú̦³ (11, 60, 61) ³o¤@²Õ¦X¡A¤S§ä¨ì 11 ©M 61 ¨â¯À¼Æ¤F¡C
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·íµM¤£·|³o¼Ë²³æ§a¡CY¨ú u = 9 ¤Î v = 8¡A§Ú̦³ u2 - v2 = 17¡A³o¬O¤@¯À¼Æ¡A¦ý u2 + v2 = 145¡A¤£¬O¯À¼Æ¡C
¨ä¹ê§Ṳ́F¸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§Ú̦Ҽ{¨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) ªº¼ÆȬO¬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§Ú̦³
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§Ú̺٧@Åp¥Í²¦¤ó¼Æ (Twin Pythagoras Numbers)¡C