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	¡m¤E³¹ºâ³N¡n¹ï©óªìµ¥¼Æ¾Ç±Ð¾Çªº»ù­È¬O¤£®e©¿µøªº¡C¤¤°ê¥j¥Nªº¼Æ¾Ç¶Ç²Î´N¬O³þ°ò©ó³o¨÷ÀÀÃD¦³½ì½T¹ê¡A³y³Nºë²Àu¬üªº²ßÃD¶°¡C(1) ´N¥H¨÷¥½ªº¡m¤ÄªÑ³¹¡n¦Ó¨¥¡AÁöµM²{¦sª©¥»ªº´¡¹Ï³£¬O«á¥@ª`®a­Ì²K¸Éªº¡A¡m¤E³¹¡n­ì¥»³ëºâ©ó¹Ïªº¨Æ¹êÓý¬OµL¥i¸mºÃªº¡C¨Ò¦p³Ì°ò¥»ªº¤ÄªÑ©w²z¡Gª½¤
T¨¤§Î(¤S
ºÙ¤ÄªÑ§Î(2) )©¶¤èµ¥©ó¤ÄªÑ¥­¤è©M¡Ga2 + b2 = c2, ¨ä¯u½T©Ê¤ñ¸û¤U¦C¨â¹Ï«K¤@¥Ø¤FµM¡G











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		³oºØ´X¦ó¦¡ªº¥N¼Æ (geometric algebra) ¤£¶È¦bªìµ¥¼Æ¾Ç±Ð¾Ç¤W°_µÛ­«­nªº±Òµo§@¥Î¡A´N¬O¹ï©ó²ßºD¥Îµ¥¸¹¥N¼Æ¾÷±ñ©Ê¸ÑÃDªº¤H¡A°¸µM¤]±a¨Ó¹y®©¡u¦¹¤¤¦³¯u·N¡vªº¼Ö½ì¡C¥»¤å¦®¦b¤¶²Ð²M©u¼Æ¾Ç®a¼BÂUµ{(3) ¦p¦ó¥©§®¦a«ö¡m¤ÄªÑ³¹¡nªº¶Ç²Îºc§@³sÄò¼Æ¤ÄªÑ§Î(¨âµuÃä¬Û®t1ªº¾ã¼Æª½¤T¨¤§Î
)¡A¨Ã§@¸É¥R©M±À¼s¡C


¤G.	¾ã¼Æ¤ÄªÑ§Îªººc§@

		¾ã¼Æ¤ÄªÑ§Îªººc§@¦b±Ð¾Ç©MÀÀÃDªº¥\¥Î¬O©úÅ㤣¹Lªº¡Cªì¾ÇªÌ³£¼ôª¾ (3, 4, 5), (5, 12, 13), (8, 15, 17) µ¥¾ã¼Æ¤ÄªÑ§Î¡C¼BÀ²µ¹¡m¤ÄªÑ³¹¡n²Ä¤Q¤GÃD(4) ªºµùÄÀ«K´¦¥Ü¤F¤@¯ë¾ã¼Æ¤ÄªÑ§Îªº§@ªk¡C¹Ï¤G®i¥Ü¤ÄªÑ§Îªº©¶¤è¡A¨ä¥ª¤W¤Î¥k¤U¨âºÝ¤À§O¬°¤Ä¤è©MªÑ¤è¡C«ö¤ÄªÑ©w²z¡A³o¨â¥¿¤è§Î
ªº­±¿n©Mµ¥©ó©¶¤è¡C¦]¦¹¥k¤W¤Î¥ª¤U¨âºÝ³Q¿òº|ªº¯x§Î (Ãäªø¦U¬° c - a ¤Î c - b) ­±¿n©M«ê¦nµ¥©ó­«Å|©ó¤¤¥¡(Ãäªø¬° a + b - c) ªº¥¿¤è§Î­±¿n¡C³o´N¬OùÚµ¥¦¡(1)¡C¦P²z¡AùÚµ¥¦¡(2)¥i¥H«ö¹Ï¤T±o¥X¡C
			(1)		2(c - a)(c - b) = (a + b - c)2
			(2)		2(c + a)(c + b) = (a + b + c)2
















				¹Ï¤G						¹Ï¤T						
		¡m¤ÄªÑ³¹¡n²Ä¤Q¤GÃD´N¬O«öµÛùÚµ¥¦¡(1)¨Ó¨D¸Ñ¤wª¾¤Ä©¶®t c - a ©MªÑ©¶®t c - b ªºª½¤T¨¤§Î¡C­Y¦¹¬°¤wª¾¡A«h«ö(1)¦¡¶}¤è±o©¶©M®t¡G
		a + b - c = ¡C
¦]¦¹
		a = (a + b - c) + (c - b) =  + (c - b)
(3)		b = (a + b - c) + (c - a) =  + (c - a)
		c = (a + b - c) + (c - a) + (c - b)
		  =  + (c - a) + (c - b)

		ºc§@¾ã¼Æ¤ÄªÑ§Î¡A¥i¥H¨ú¥ô·N¨â¾ã¼Æ  p¡Bq¡A ¥O   c - a  = 2p2¡F            c - b  =  q2¡A§Y±o  a + b - c 
= 2pq¡C±q¦¹±o¤ÄªÑ§Î¤T¾ã¼ÆÃä

			a = 2pq + q2
(4)			b = 2p2 + 2pq
			c = 2p2 + 2pq + q2

­Y p¡Bq ¤¬¯À¡A«h¾ã¼Æ¤ÄªÑ§Î¤TÃ䤣±a¤½¦]¤l¡AºÙ§@¥»­ì (primitive) ¤ÄªÑ§Î¡C


¤T. ¼BÂUµ{ªº³sÄò¼Æ¤ÄªÑ§Îºc§@

		·íµM¡A«ö(4)¦¡¡A³sÄò¼Æ¤ÄªÑ§Îªººc§@Âkµ²¨ì¨D¸Ñ¤£©w¤èµ{                 2p2 - q2 = 1  ªº°ÝÃD¤W¡C(5)  ¼BÂUµ{«oª½±µ±qùÚµ¥¦¡(1)¡B(2)¤J¤â¡Aµ¹¥X¤@­Ó­¡¥N¤èªk¡A«ö¤@¤ÄªÑ§Î (a¡Ab¡Ac) ºc§@¸û¤jªº¤ÄªÑ§Î (a'¡Ab'¡Ac')¡A¨Ï±o¤ÄªÑ®t¤£ÅÜ¡G b' - a' = b - a¡C¨äªk¤D¬O¨ú  c' - a
' = c + b¡A¤Î  c' - b' = c + a¡C «ö(1)¡B(2)¨â¦¡¡A±oª¾  a' + b' - c' = a + b + c ,  ±q¦Ó (6)
			a' = 2a + b + 2c
(5)			b' = a + 2b + 2c
			c' = 2a + 2b + 3c

¼B¤ó±q³Ì¤pªº¾ã¼Æ¤ÄªÑ§Î (3, 4, 5)¶}©l¡A«ö(5)¦¡­¡¥N±o³sÄò¼Æ¤ÄªÑ§Î¦p¤U¡G

(6)		(3, 4, 5), (20, 21, 29), (119, 120, 169), (696, 697, 985), ...


¥|. ¼B¤óºc§@ªº§¹³Æ©Ê

		§Ú­Ì¦ÛµM­n°Ý¡G³o­Ó­¡¥Nµ{§Ç(5)¬O§_µ¹¥X©Ò¦³³sÄò¼Æ¤ÄªÑ§Î¡C¼B¤ó¦ü¥G¨S¦³°Q½×³o­Ó°ÝÃD¡C¯÷¸É­z©ó¦¹¡C®e©ö¹îı­¡¥N(5)¬O¥i°fªº¡G±q¤@¾ã¼Æ¤ÄªÑ§Îºc§@¤ÄªÑ®t¤£Åܪº¸û¤p¾ã¼Æ¤ÄªÑ§Î¡C§Y¬O»¡
			a = 2a' + b' - 2c'
(7)			b = a' + 2b' - 2c'
			c = -2a' - 2b' + 3c'
		«ö¤T¨¤¤£µ¥¦¡  a' + b' > c'  ®e©ö±oª¾  c < c'¡C¦]¦¹ (a, b, c) ¬°¸û¤p¤ÄªÑ§Î¡CµM¦Ó³o­Ó¤Ï­¡¥Nµ{§Ç (7) Á`¤£¯à±q¤@¾ã¼Æ¤ÄªÑ§Î¥ÃµL¥ð¤î¦aµ¹¥X»¼´îªº¾ã¼Æ¤ÄªÑ§Î¡CÃöÁä¦b©ó³ÌµuÃä  a = 2a' + b' - 2c'  ¥²¶·¬°¥¿¾ã¼Æ¡C«ö¤ÄªÑ©w²z¡A®e©ö±q  2a' + b' > 2c'  ±À±o  4a' > 3b' ¡
A¥ç§Y  a' > 3(b' - a')¡C ¦]¦¹¡A«ö­¡¥N (7)¡A¥ô¦ó¤ÄªÑ®t¬° d ªº¾ã¼Æ¤ÄªÑ§Î¥²ÁÙ­ì¨ì¤Ä¤£¤j©ó 3d ªº(·¥¤p)¾ã¼Æ¤ÄªÑ§Î¡C¹ï©ó³sÄò¼Æ¤ÄªÑ§Î¦Ó¨¥¡A d = 1¡C¤Ä¤£¤j©ó 3 ªº(³sÄò)¾ã¼Æ¤ÄªÑ§Î«ê¦n¥u¦³ (3, 4, 5) ¤@­Ó¡C³o´N»¡©ú¼B¤óªººc§@ (5) ¯uªºµ¹¥X¤F©Ò¦³³sÄò¼Æ¤ÄªÑ§Î¡C


¤­. ¤ÄªÑ®t¬°©w¼Æªº¾ã¼Æ¤ÄªÑ§Î

		¼BÂUµ{ªº­¡¥N (5) ·íµM¤£©ë­­©ó³sÄò¼Æ¤ÄªÑ§Îªººc§@¡C¹ï©ó¤ÄªÑ®t¬°©w¼Æ d > 1 ¤§¥»­ì¤ÄªÑ§Î¡A«ö (7) ¥²ÁÙ­ì¨ì³ÌµuÃä¤p©ó 3d ªº¥»­ì¾ã¼Æ¤ÄªÑ§Î¡C(7)¤µ±ý¨D¤ÄªÑ®t¤p©ó 10 ¤§¥»­ì¾ã¼Æ¤ÄªÑ§Î¡C«ö (4) ¦¡ª¾¥»­ì¤ÄªÑ§Î¤§¤ÄªÑ®t¥²¬°©_¼Æ¡C¦]¦¹¤ÄªÑ®t¤p©ó 10 ¤§¥»­ì¤ÄªÑ§Î¥²ÁÙ­ì¨ì³Ì
µuÃä¤p©ó 27 ¤§¥»­ì¤ÄªÑ§Î¡C«ö (4) ¦¡Åçºâ¦¹Ãþ¾ã¼Æ¤ÄªÑ§Î¥u¦³¤U¦C¥|ºØ¡G

p
q
(a, b, c)
|b - a|

1
1 
(3, 4, 5)
1

2
1
(5, 12, 13)
7

1
2
(15, 8, 17)
7

2
3
(21, 20, 29)
1


«ö¦¹¥i¥H±À±o¤U¦Cµ²½×¡G
(¤@)	¤£¦s¦b¤ÄªÑ®t¬° 3¡B5¡B9 ªº¥»­ì¤ÄªÑ§Î¡C(8) 
(¤G)	¤ÄªÑ®t¬° 7 ªº¥»­ì¤ÄªÑ§Î¦@¦³¨â²Õ¡A¤À§O¥Ñ(5, 12, 13),(8, 15, 17)
		«ö (5) ¦¡­¡¥N±o¥X¡G

			(5, 12, 13), (48, 55, 73), (297, 304, 425), ...
			(8, 15, 17), (65, 72, 97), (396, 403, 565), ...

¼B¤ó´N¬O§Q¥Î³o­Ó¤èªk±o¥X¤ÄªÑ®t¬° 7 ªº¥»­ì¤ÄªÑ§Î¡A¨Ã¥B¥©§®¦a¹B¥Î¤U¦C¨â¹Ï±o¥XªºùÚµ¥¦¡ (8)¡B(9) ¨Ó¨D¸Ñ¤£©w¤èµ{  2x2 + 49 = z2¡C

(8)			(a + b)2 = 2c2 - (b - a)2 
(9)			(2c - a - b)2 = (b - a)2 + 2(a + b - c)2



















					¹Ï¥|					¹Ï¤­
		
		§Ú­Ì­À®Ä¼B¤óªº¤èªk§Q¥Î³o¨â«hùÚµ¥¦¡ (8), (9) ¨Ó¨D¸Ñ¨â¹D¦³½ìªº¼Æ½×°ÝÃD¡C


¤». ¤T¨¤§Î¼Æµ¥©ó¥­¤è¼Æ°ÝÃD

		¤T¨¤§Î¼Æ (triangular number) ¬O«ü§Î¦p 1 + 2 + 3 +... + n ªº³sÄò¾ã¼Æ©M¡A¦¹§Y 1, 3, 6, 10, 15, 21, 28, 36, ...¡C¤µ°Ý°£ 1 ©M 36 ¥~¡A¬O§_©|¦³¼Æ­È¬°¥­¤èªº¤T¨¤§Î¼Æ¡CÅãµM²Ä n ­Ó¤T¨¤§Î¼Æ¥i¥Hµø¬°µuÃ䬰³sÄò¼Æ n, n + 1 ªº¤ÄªÑ§Î­±¿n¡C(9)


















¹Ï¤»

		­Y²Ä n ­Ó¤T¨¤§Î¬°¥­¤è¼Æ  m2¡A «ö¹Ï¥|¡A ¥O   a = n,    b = n + 1           ±o  1 + 8m2 = 1 + 2(2m)2  ¬°¥­¤è¼Æ¡C¤ñ¸ûùÚµ¥¦¡ (9) ª¾¦s¦b¤@³sÄò¤ÄªÑ§Î          (a, b, c) ¨Ï±o  2m = a + b - c¡F  2n + 1 = 2c - a - b¡C
§Y   n = a - (c - 1)¡F m = c - b¡C ¤µ«ö (6) ¦¡ªº³sÄò¼Æ¤ÄªÑ§Î±o 
			(n, m)  =  (1, 1),  (8, 6),  (49, 35), (288, 204), ...
¨ä¾l¥i«ö
			n' = 3n + 4m + 1
			m' = 2n + 3m + 1
­¡¥N¦Ó±o¡C


¤C.	¤T¨¤§Î¼Æ¬°¥t¤@¤T¨¤§Î¨â­¿°ÝÃD

		³]²Ä n ­Ó¤T¨¤§Î¼Æ¬°²Ä m ­Ó¤T¨¤§Î¼Æ¤§¨â­¿¡C§Y










¹Ï¤C

	«ö¹Ï¤K¡AÃäªø¬° 2m + 1 ¤§¥¿¤è§Î¥i¥H­«·s³¯¦C¦p¹Ï¤E¡G






					



					¹Ï¤K					¹Ï¤E

¥Ñ¬O±oª¾  (n, n + 1, 2m + 1)  ¬°³sÄò¼Æ¤ÄªÑ§Î¡C¤ñ¸û (6) ¦¡±o

		(n, m) =  (3, 2), (20, 14), (119, 84), (696, 492), ...

¨ä¾l¥i«ö
				n' = 3n + 4m + 3
				m' = 2n + 3m + 2
­¡¥N¦Ó±o¡C

¤K. ²ßÃD

1.	ø¹Ï»¡©ú¨â³sÄò¤T¨¤§Î¼Æ¤§©M¥²¬°¥­¤è¼Æ¡A¨Ã¨D¥|³sÄò¤T¨¤§Î¼Æ¨Ï¨ä©M¥ç¬°¥­¤è¼Æ¡C

2.	¥Òµó¬ù¦³©Ð«Î¤T¦Ê¶¡¡A±i§g¦í¦v¦b¥Òµó©_¼ÆªùµP¤@°¼¡C¨ä¥ª¾F(¦ÛµóÀY°_)ªùµP¸¹¼ÆÁ`©M«ê¦nµ¥©ó¨ä¥k«Z (¨´µó§À¤î) ªùµP¸¹¼ÆÁ`©M¡C°Ý¥Òµó¦@¦³©Ð«Î­Y¤z¡H±i§gªùµP´X¦ó¡H

3.	¤Aµó¬ù¦³©Ð«Î¤»¦Ê¶¡¡A§õ§g¦í¦v¦b¤Aµó°¸¼ÆªùµP¤@°¼¡C¨ä¥ª¾F(¦ÛµóÀY°_)ªùµP¸¹¼ÆÁ`©M«ê¦nµ¥©ó¨ä¥k«Z (¨´µó§À¤î) ªùµP¸¹¼ÆÁ`©M¡C°Ý¤Aµó¦@¦³©Ð«Î­Y¤z¡H±i§gªùµP´X¦ó¡H

¤E. ²ßÃD (¤£©w¤èµ{  1 + 3n2 = m2 )

1.	³]¦³¤ÄªÑ§Î (a. b. c)¡C¸ÕÃÒùÚµ¥¦¡
				(c + b) (5c - 4a + 3b) = (2c - a + 2b)2
	ø¹Ïµý©ú³Ì¨Î¡C

2.	³]¦³¤ÄªÑ§Î (a, b, c)¡A¨D§@¤ÄªÑ§Î (a', b', c')¡A ¨Ï
				c' - a' = c + b	 ¤Î  c' + a' = 5c - 4a + 3b
	µý©ú 2a - c ¤§µ´¹ï­È¤£ÅÜ¡G  2a' - c' =  - (2a - c)

3.	±q³Ì¤p¤ÄªÑ§Î (3, 4, 5) ¶}©l¡A«ö«eÃDºc§@©Ò¦³  2a - c = 1  ¤§¾ã¼Æ¤ÄªÑ§Î¡C

4.	µý©úÃD (1) ¤§ùÚµ¥¦¡¥i¥H§ï¼g¦p¤U¡G
				(2a - c)2 + 3(a + 2b - 2c)2  =  (4c - 2a - 3b)2

5.	§Q¥ÎÃD (3) ¤§¾ã¼Æ¤ÄªÑ§Î¨D¸Ñ¤£©w¤èµ{  1 + 3n2 = m2  ±o
				(n, m)  =  (1, 2), (4, 7), (15, 26), (56, 97), ...
	¸Õ»¡©ú¨ä­¡¥Nµ{§Ç¡A¨Ãµý©ú¨ä§¹³Æ©Ê¡C


¤Q. ²ßÃD (¤T±×¨D¿n)

			¯³¤E»à¡m¼Æ®Ñ¤E³¹.¨÷¤­¡n¦³¡u¤T±×¨D¿n¡vÃD¡A³]¤T¨¤§ÎÃäªø 13, 14, 15¡A¨D±o­±¿n 84¡C ¤@¯ë¦Ó¨¥¡A¤T¨¤§Î(¤£¥²¬°¤ÄªÑ§Î)µ¹©wÃäªø  a, b, c, ¨ä­±¿n¥i«ö¤U¦C¤½¦¡¨D±o:(10)
						­±¿n  =  
	¤µ±ý¨D¤TÃäªø«×¬°³sÄò¾ã¼Æ¡A­±¿n¥ç¬°¾ã¼Æ¤§¤T¨¤§Î¡C
	1.	³]¤TÃäªø«×¬° b - 1, b, b + 1¡C¨Dµý b ¥²¬°°¸¼Æ¡C
	2.	¥O b = 2m¡C¨Dµý m2 - 1 ¬°¤@¥­¤è¼Æ¤§¤T­¿¡C
	3.	«ö¤W¸`¤£©w¤èµ{  1 + 3n2 = m2  ¤§³q¸Ñ±o¤U¦C¤TÃ䬰³sÄò¼Æ¡A­±¿n¬°¾ã¼Æ¤§¤T¨¤§Î

m
(b - 1, b, b + 1)
­±¿n

2
(3, 4, 5)
6

7
(13, 14, 15)
84

26
(51, 52, 53)
1170


			¸Õ»¡©ú¨ä­¡¥Nµ{§Ç¡A¨Ãµý©ú¨ä§¹³Æ©Ê¡C




µùÄÀ¡G

(1)	°ÑÊ÷¥Õ©|®¤¡G¡m¤E³¹ºâ³Nª`ÄÀ¡n¡A¬ì¾Ç¥Xª©ªÀ 1983¡A¤Î§d¤å«T¥D½s¡m¤E³¹ºâ³N»P¼BÀ²¡n¡A¥_®v¤j¥Xª©ªÀ 1982¡C ­^¤å°Q½×¡m¤ÄªÑ³¹¡nªº±MµÛ¦³ F.J.Swetz, Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China. Penn. State Univ. Press, 1977, ¤
Î LAM Lay-Yong & SHEN Kangsheng, Right-Angled Triangles in Ancient China, Arch. Hist. Exact Sciences, 30(1984)87-112.

(2)	ª½¨¤¨â¾FÃäµuªÌ¬°¤Ä¡AªøªÌ¬°ªÑ¡A¤À§O°O§@ a, b¡Cª½¨¤¹ïÃ䬰©¶¡A°O§@ c¡C¤ÄªÑ§Î a, b ¨âÃäªøµu¤§¤À¡A¤£©y¹L¥÷©ëªd¡CÄ´¦p (4) ¦¡¤¤ a, b ¤§¬Û¹ïªøµu¡Aµø¥G p, q ¤§¿ï¾Ü¦Ó²§¡C

(3)	¥»¤å°Q½×¼B¤ó¹ï³sÄò¼Æ¤ÄªÑ§Îªººc§@¡A¨ú§÷©ó¿úÄ_Úz¡m¤¤°ê¼Æ¾Ç¤¤¤§¾ã¼Æ¤ÄªÑ§Î¬ã¨s¡n¦¬¤J¡m¿úÄ_Úz¬ì¾Ç¥v½×¤å¿ï¶°¡n¡A¬ì¾Ç¥Xª©ªÀ¡A1983¡A­¶287-303¡C¸Ó¤å­¶303®i¥Ü (3)¦¡¥ªÃäÀ³¬° a' + b' - c'¡C¤S«ö¬x¸U¥Í¡G¡m¦P¤åÀ]ºâ¾Ç±Ð²ß§õµ½Äõ¡n(¦¬¤J·¨»AµØ¡B¶À¤@¹A¥D½s¡G¡mªñ¥N¤¤°ê¬ì§
Þ¥v½×¶°¡n¡A²MµØ¾ú¥v©Ò¥Xª©¡A1991)­¶237¡G¡u¼BÂUµ{...´¿¥ô¤W®ü¼s¤è¨¥À]ºâ¾Ç±Ð²ß(1873)¡A«á­Ý¥D«ù(¤W®ü)¨D§Ó®Ñ°|ºâ¾Ç¬ì(1875)¡C¦b1893¦~°h¥ð«á¡A¥L±N¦h¦~©Ò¥X¸ÕÃD¤Î¸Ñµª½s¦¨¡m²©ö±gºâ½Z¡n(1899)¡A¦b¼Æ½×ªº¤£©w¤ÀªR¬ã¨s°µ¥X¤@ÂI¦¨ÁZ¡v¡C¦~¨Ó»X¬x±Ð±Â´fÃؤ¤°ê¼Æ¾Ç¥v½×¤å¦h½g¡AÀ
ò¯q¨}¦h¡AÂÔ¦¹»ïÁ¡C

(4)	¡m¤ÄªÑ³¹¡n²Ä¤Q¤GÃD¡G¤µ¦³¤á¤£ª¾°ª¼s¡A¬ñ¤£ª¾ªøµu¡C¾î¤§¤£¥X¥|¤Ø¡A±q¤§¤£¥X¤G¤Ø¡A¨¸¤§¾A¥X¡C°Ý¤á°ª¼s×À¦U´X¦ó¡H

(5)	¦è¤è¾Ç¬É¹ï©ó¦¹°ÝÃDªº¬ã¨s¡B°Ñ¦Ò L.E.Dickson, History of the Theory of Numbers, vol.II,(1920), Chelsea reprint, pp.181-184.

(6)     (5)¦¡ªºa', b', c' ©T¥i¨D¸ÑÁp¥ß¤èµ{¦Ó±o¡AµM¥ç±o«ö¹Ï¤G¯Á¥X¡C±N¹Ï¤G¤§ªø«× a, b, c§ï§@ a', b', c', «h
			a'	= (a' + b'- c') + (c' - b')
	  		= (a + b + c) + (c + a)
					= 2a + b + 2c
	¦P²z±o b', c'¡C¤U­±(7)¦¡¤§ a, b, c ¥ç¥i«ö¹Ï¤TŪ¥X¡C

(7)	³ÌµuÃ䬰 3d¡A¤ÄªÑ®t¬° d ¤§¤ÄªÑ§Î¥²¬° (3d, 4d, 5d)¡C d>1 ®É¦¹«D¥»­ì¤ÄªÑ§Î¡C

(8)	¤£©w¤èµ{  2p2 - q2 = 3, 5, 9 µL¸Ñ¤§¨Æ¹ê¡A¥ç¥i«ö 2¬°¼Ò 3(©Î¼Ò 5)¤§¤G¦¸«D³Ñ¾l (quadratic nonresidue) ±À¾É±o¤§¡C

(9)	¤U­±°²³]¹Ï¤»¥kºÝ¤§¤ÄªÑ§Î­±¿n¬°¥­¤è¼Æ¡A¨ä©¶ªø¥²¤£¯à¬°¾ã¼Æ¡A¦]¬°¶O     º¸º¿ (Pierre de Fermat, 1601-1665) ´¿§Q¥ÎµÛ¦WªºµL½a¤U­°ªk (method of infinite descent) µý©ú¾ã¼Æ¤ÄªÑ§Î¤§­±¿n¥²¤£¯à¬°¥­¤è¼Æ¡A¥»¤å²Ä¥|¸`ªºµý©ú¡A´N»P¦¹µL½a¤U­°ªk¦³¬Û¦P¦®½ì¡C

(10)	±q¦¹­±¿n¤½¦¡¥ç¥i±À±oµÛ¦Wªº®ü­Û¤½¦¡ (Heron formula)
­±¿n  =  ,
         ¨ä¤¤  ¬°¤T¨¤§Î¤§¥b©P¬É¡C