¥u¦³1¡B2¡B3 ªº¼Æ¦C - ´y­z¼Æ¦C

¬ü°ê¼Æ¾Ç®a±d­³ (John Horton Conway 1937- )

(·Ó¤ù¨ú¦Û¡uThe MacTutor History of Mathematics Achieve¡vhttp://www-gap.dcs.st-and.ac.uk/~history/ )

¡@

¬Ý¬Ý³o¼Æ¦C§a

1¡B11¡B21¡B1211¡B111221¡B312211¡B13112221¡B1113213211¡B31131211131221¡B...... (OEIS A005150)

³o¬O¤@©_©Çªº¼Æ¦C¡A¥u¦³¤T­Ó¼Æ¦r - 1¡B2 ©M 3¡C

³o§Ú­ÌºÙ¬°´y­z¼Æ¦C (Descriptive Sequence) ©Î ¦Û´y¼Æ¦C (Self-descriptive Sequence) ©Î »y¨¥¼Æ¦C (Look And Say Sequence)¡GÁ`¤§¦W¦r¦h¦h¡C

¡@

°_©l¬° 1¡A²Ä¤G¶µ¬O´y­z²Ä¤@¶µªºµ²ªG¡A§Y¦³ 1 ­Ó 1¡F²Ä¤T¶µ¬O´y­z²Ä¤G¶µªºµ²ªG¡A§Y¦³ 2 ­Ó 1¡F²Ä¥|¶µ¬O´y­z²Ä¤T¶µªºµ²ªG¡A§Y¦³ 1 ­Ó 2 ©M 1 ­Ó 1¡G¦p¦¹¤U¥h¡C

¡@

§Ú­Ì¤£Ãø§ä¨ì¤T­Óµ²½×¡G

¼Æ¦C¬Oµo´²ªº ¡A¦Ó¥B»á¬°¡u§Ö¤j¡v¡C

¼Æ¦C¤¤©Ò¦³¼Æ­Èªº­Ó¦ìùÚ¬° 1¡C

¼Æ¦C¤¤¦U¼Æ­È¥u¦³¤T­Ó¼Æ¦r §Y 1¡B2 ©M 3 ©Ò²Õ¦¨¡C

¡@

¹ï©ó¥H¤W²Ä¤@ÂI¡A¤£¥Î¦h§@¸ÑÄÀ¡A¤å¦r´y­z¬O¶O®Éªº¡A­þ¦³¼Æ¦rªºÂ²¼ä¡I

²Ä¤GÂI¡A¤]«Ü¦ÛµM¡A¼Æ¦C³Ì«áÁ`¬O´y­z¦h¤Ö­Ó 1¡A¬G·s¤@¶µ¼Æ¦ÛµM¬O¥H 1 ¬°­Ó¦ì¤F¡C

¡@

²Ä¤TÂI«h¥Î¦h¤@¤BÂI®É¶¡¨Ó¸ÑÄÀ§a¡I­Y¯uªº¥X²{ 4 (©Î¥H¤Wªº¼Æ¦r)¡A¥²µM¦b«e¤@¶µ¥X²{¤F 4 (©Î¥H¤Wªº¼Æ¦r) ©Î¬O³sÄò¥X²{¤F 4­Ó (©Î§ó¦h­Ó) ¬Û¦Pªº¼Æ¦r¡C·íµM²Ä¤@­Ó¥i¯à©Ê³Ì²×±À¾Éªº·½ÀY¤]¬O²Ä¤G­Ó­ì¦]¡A¦]¬°¶}©l¬° 1 ¡A¤£·|µL¬G¥X²{ 4 ©Î¥H¤Wªº¼Æ¦r¡C¦ý°ÝÃD¬O·|¤£·|¦³ 4­Ó (©Î§ó¦h­Ó) ¬Û¦Pªº¼Æ¦r³sÄò¥X²{©O¡H¨º´NÅý§Ú­Ì¥ý¬Ý¬Ý¦³ 4 ­Ó¬Û¦P¼Æ¦rªº±¡ªp§a¡I

°²³]¥X²{¤F³sÄò 4 ­Ó¬Û¦Pªº¼Æ¦r A¡A§Y ......AAAA...... ¡A¦¡¤¤ ...... ¥Nªí¨ä¥L¼Æ¦r¡A¤U¦P¡C­Y§Ú­Ì¦Û³Ì¥ª ¼Æ¦r¶}©l¥H¼Æ¦r¨â¨â¤@²Õ¡A¨º¥u¦³¨â­Ó¥i¯à©Ê¡A¤@¬O³o¥|­Ó A ¤À¦C¨â²Õ¡A§Y ...... (AA)(AA) ...... ©Î¸m©ó¤T²Õ¡A§Y ...... (XA)(AA)(AY)......¡C

¥ý½Í²Ä¤@­Ó¥i¯à©Ê¡A§Y¦A«e¤@¶µ¥X²{¤F A ­Ó A ©M A ­Ó A ¡A¦ý§Ú­Ì¤£·|§â³o¨â²Õ¬Û¦Pªº¼Æ¦r¤À¶}´y­z¡A¥u·|»¡¤@¥y¦³ 2A ­Ó A ¡A¬G¥X²{³o±¡ªp¬O¤£¥i¯àªº¡C¦Ó²Ä¤G­Ó¥i¯à©Ê¤S¦p¦ó¡A§Y¦A«e¤@¶µ¥X²{¤F X ­Ó A ¡BA ­Ó A ©M A ­Ó Y¡A§Ú­Ì¦P¼Ë¦a§â (X+A) ­Ó A ¤À¶}³ø¼Æ¡A¬O¤£¥i¯àªº¡C¦]¦¹¥X²{ 4 ©Î¥H¤Wªº¼Æ¦r¬O¤£¥i¯àªº¡A°£«D³o­Ó 4 ©Î¥H¤Wªº¼Æ¦r¬O»P¥Í­Ñ¨Óªº§a¡C

¦Ü©ó 0 ©O¡H³o§ó®e©ö¸Ñ»¡¤F¡A°_ªì¤£´¿¦³¹s¡A¬G¤£·|»¡¥X¦³­Y¤z¹s³o¦^¨Æ¡F¦Ü©ó¦³¡u¹s­Ó¬Y¼Æ¡v©O¡A§ó¬O»¡¤£±o¡C¬O¬G§Ú­Ì¤]¤£·|¬Ý¨ì¼Æ¦C¤¤¦³¹s¤F¡C

¡@

­Y§Ú­Ì¶}©lªº¤£¬O 1 ¡A¦Ó¬O¨ä¥L¥Ñ 2 ¦Ü 9 ªº¼Æ¦r¤S¦p¦ó¡A¬Ý¬Ý¤Uªí¡G

2
2, 12, 1112, 3112, 132112, 1113122112, 311311222112, ......
3
3, 13, 1113, 3113, 132113, 1113122113, 311311222113, ......
4
4, 14, 1114, 3114, 132114, 1113122114, 311311222114, ......
d
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, ......

¤Wªí¤¤ d ¬° 2 ¦Ü 9 ¥ô¦ó¤@¼Æ¡C§Ú­Ì¤£Ãøµo²{·í¤¤ªº¦@³q¡C

¡@

±d­³±`¼Æ

­ì¨Ó³o¨Ç´y­z¼Æ¦Cªº¼Æ­È¬O·|º¥ªñ¤@¥N¼Æ¦¡ T(n) = ¡A

¦¡¤¤ªº = 1.567......¡B= 1.814...... ¡B = 1.3035772690342963912570991121525518907307025046594...... ¡A³o ¤SºÙ¬°±d­³±`¼Æ (Conway's Constant) (OEIS A014715)¡A¦Ó³o±`¼Æ³º¬O

ªº°ß¤@¥¿¹ê®Ú¡C

¡@

³o¼Æ¦C¤¤¦³¯À¼Æ¶Ü

¦p¬ã¨s¥H 1 ¬°­ºªº¨º±ø´y­z¼Æ¦C¡A§Ú­Ì§ä¨ì·í¤¤¦³ 11¡B 312211¡B 13112221³o¤T­Ó¯À¼Æ¡A·íµM³°Äò¦³¨Óªº¡C ¨º¯À¼Æ§Ú­ÌºÙ¤§¬°´y­z¯À¼Æ (Descriptive Prime)¡C

·í¤¤¥ç¦³¼Æ¾Ç®a§ä´M³sÄòªº´y­z¯À¼Æ (Consecutive Descriptive Primes)¡A©Î·|§_¥X²{´y­z ¯À¼Æ¼ÆÃì (Chain of Descriptive Primes) ©O¡H¨ä¤¤±d¯Ç¥d (G. L. Honaker, Jr.) «K§ä¨ì 373, 131713, 111311171113, 3113331173113 ³o¤@¼Æ¦C¡A¥i±¤¨ä¤U¤@¶µ¬O¤@¦X¼Æ¡C¦Ó¦b 2003¦~¡A¨½¥±µØ (Carlos B. Rivera, 1950 - )¡B°ò«ä (Mike Keith 1955 - ) ©M µØ¹F¡DµÎ¥§¹F (Walter Schneider) §ä¨ì¤@ªø¤C¶µªº´y­z¼Æ¦C©l¦Û 19972667609¡C·|§_¦³§óªøªº´y­z¯À¼Æ¼ÆÃì¡H§Ú¬Û«H·|¦³ªº¡C

¡@

°Ñ¦Ò¤åÄm¤Îºô§}¡G

Rivera, C. "Problems & Puzzles: Puzzle 036 - Sequences of descriptive primes." http://www.primepuzzles.net/puzzles/puzz_036.htm

Schneider, Walter. "Descriptive Primes" http://www.wschnei.de/digit-related-numbers/descriptive-primes.html

Weisstein, E. W. "Look And Say Sequence." From MathWorld http://mathworld.wolfram.com/LookandSaySequence.html .

½Í²»¬f. "®¶Àú»P¼Éº¦ªº»y¨¥¼Æ¦C" ¦Û ¼Ö¦b¨ä¤¤ªº¼Æ¾Ç , ¥_¨Ê : ¬ì¾Ç¥Xª©ªÀ , p. 333-335 , 2005

Free Web Hosting