ùÚµ¥¦¡ªºÁp·Q
¦³¤H±q§¹¥þ¥¤èùÚµ¥¦¡¤¤§ä¨ìÆF·P¡A°Q¨Ó¨âÃþ¯À¼Æ¡G¥dùº¸¯À¼Æ (Carol Prime) ¤Î³Í¥§¨È¯À¼Æ (Kynea Prime)
²³©Ò©Pª¾§¹¥þ¥¤èùÚµ¥¦¡¬O«ü (a+b)2 = a2 + 2ab + b2 ¤Î (a-b)2 = a2 - 2ab + b2
Y¨ú a = 2x ¡Ab = 1¡A(a+b)2 - 2 «K¬O³Í¥§¨È¯À¼Æ¡A¦Ó (a-b)2 - 2 «h¬O¥dùº¸¯À¼Æ¡C
¥dùº¸¯À¼Æ©M³Í¥§¨È¯À¼Æªº¦W¦r¬O¦Û¨â¤H Carol G. Kirnon ¤Î Kynea R. Griffith ªº¦W¦r¡C¦Ü©ó¬°¦ó¥H¤Wz¤G¤Hªº¦W¦r§@¬°¸Ó¯À¼ÆÃþ§O¤§¦W¡Aì¦]¬O³Ì¦¬ã¨s³o¨âÃþ¯À¼Æªº¾ÇªÌ¦ã¤å¯Ãº¸ (Cletus Emmanuel) ´À³oªù¯À¼Æ«a¤W¥LªºªB¤Íªº¦W¦r¡C
³Í¥§¨È¯À¼Æ¥ç¥i¥H 4x + 2x+1 - 1 ªº«¬¦¡®Ñ¼g¡A¦P²z¥dùº¸¯À¼Æ¤]¦P¥i¥H 4x - 2x+1 - 1 ªº«¬¦¡®Ñ¼g¡C
Y§Ú̧⥤è§ï¬°¤T¦¸¤è¡A«h¦³¤F¥H 3 ¬°©³ªº³Í¥§¨È¯À¼Æ©Î¥dùº¸¯À¼Æ¡C¾l¦¹Ãþ±À¡C
¥tY¨ú a = kx ¡A k ¥i¥H¤£µ¥©ó 2 ¡A«h¦³¤F¼s¸q³Í¥§¨È¯À¼Æ (Generalized Kynea Prime) ©Î¼s¸q¥dùº¸¯À¼Æ (Generalized Carol Prime)¡C
²{¦b¤wª¾ªº³Í¥§¨È¯À¼Æ (¥H 2 ¬°©³) ¹ê¤£¦h¡G
¥u¦³ x = 0, 1, 2, 3, 5, 8, 9, 12, 15 17, 18 21, 23, 27, 32, 51, 65, 87, 180, 242, 467, 491, 501, 507, 555, 591, 680, 800, 1070, 1650, 2813, 3281, 4217, 5153, 6287, 6365, 10088, 10367, 37035, 45873, 69312, 102435, 106380, 108888, 110615 ¦X¦@ 45 Ó¡A¨ä¤¤³Ì¤jªº¤@Ó¬° (2110615 + 1)2 - 2 ¡A¸Ó¯À¼Æ¹F 66597 ¼Æ¦ì (Digit) ¤§¦h¡A©ó 2004 ¦~¥Ñ¤å¿Õ°ì (Predrag Minovic) µo²{¡C (OEIS A093069)
¦Ó²{¦b¤wª¾ªº¥dùº¸¯À¼Æ (¥H 2 ¬°©³) ¤]¤£¦h¡G
¥u¦³ x = 2, 3, 4, 6, 7, 10, 12, 15, 18, 19, 21, 25, 27, 55, 129, 132, 159, 171, 175, 315, 324, 358, 393, 435, 786, 1459, 1707, 2923, 6462, 14289, 39012, 51637, 100224, 108127, 110953, 226749 ¦X¦@ 36 Ó¡A¥t¥~¦³¨âÓÅçºâ·í¤¤¡C¨ä¤¤³Ì¤jªº¤@Ó¬° (2226749 - 1)2 - 2 ¡A¸Ó¯À¼Æ¹F 136517 ¼Æ¦ì¤§¦h¡A©ó 2005 ¦~¬ü°ê·~¾l¼Æ¾Ç®a¥Ñ«¢ºû (Steven Harvey) µo²{¡C(OEIS A091516)
°Ñ¦Ò¤åÄm¤Îºô§}¡G
Harvey, S. "Prime Number Searches : Carol Primes and Kynea Primes" http://www.geocities.com/harvey563/Carol_Kynea.txt.
Weisstein, E. W. "Carol Number." From MathWorld. http://mathworld.wolfram.com/CarolNumber.html.
Weisstein, E. W. "Kynea Number." From MathWorld. http://mathworld.wolfram.com/KyneaNumber.html.