31=2^5-1.Euler ¤ÎÄêÍý¤«¤é¤Ï 2^30¢á1(mod 31) ¤¬ÆÀ¤é¤ì¤ë.Euler ¤ÎÄêÍý¤ò½Ð¤¹¤Þ¤Ç¤â¤Ê¤¯, 2^30=(2^5)^6¢á1(mod 31).
R ¤ò¼Â¿ôÁ´ÂΤ«¤é¤Ê¤ë½¸¹ç¤È¤¹¤ë.{(x,y);((x,y)¢ºR^2)¢Ê(y=(1-x^2)^(1/2))} ¤È {(x,y);((x,y)¢ºR^2)¢Ê(y=-x^2/2+1)} ¤Ï (0,1) ¤ÇÆ󼡤ÎÀÜ¿¨¤ò¤¹¤ë.¤·¤«¤·, Æó³¬Èùʬ¤¬¾¯¤·ÊѤï¤Ã¤Æ¤â¶¦ÍÅÀ¤ÈÀÜÀþ¤ÎÊý¸þ¤¬Æ±¤¸¤Ê¤é¸«¤¿ÌܤǤÏÆ󼡤ÎÀÜ¿¨¤ò¤·¤Æ¤¤¤ë¤«¤É¤¦¤«¤Ïʬ
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x ¤¬ 0 ¤Ç¤â -1 ¤Ç¤â¤Ê¤¤À°¿ô¤Î»þ, x^2+x+1 ¤ÏÊ¿Êý¿ô¤Ç¤Ï¤Ê¤¤.18^2+18+1=7^3.
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0 ¤ò̵¸Â·å¤ÎÆó¿Ê¿ô¤Ç 0 ¤¬Ìµ¸Â¤Ë¤¢¤ë¤È¹Í¤¨¤ë¤È, -1 ¤Ï 1 ¤Î̵¸Â¤ÎÏ¢¤Ê¤ê¤È¹Í¤¨¤ë¤«.
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x ¤È y ¤òÊ£ÁÇ¿ô¤È¤¹¤ë.ô_{k=0}^{n}(xy^k) ¤¬ n¢ª¡ç ¤Ç¼ý«¤¹¤ë»ö¤È, (x=0)¢Ë(|y|<1)¢Ë(y=1) ¤ÏƱÃͤǤ¢¤ë.
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n ¤ò 1 °Ê¾å¤Î¼«Á³¿ô¤È¤¹¤ë.(2n+1)-³Ñ·Á¤Ï¼«¸Ê¸òºµ¤ò¤·¤Æ¤âÎɤ¤»ö¤È¤¹¤ë¤È, ¤É¤ÎÆóÎǤˤⶦÍÅÀ¤¬¤¢¤ë¤è¤¦¤Ë¤Ç¤¤ë.(2n+2)-³Ñ·Á¤Ï¤¢¤ëÆóÎǤ˶¦ÍÅÀ¤¬Ìµ¤¤.
n ¤ò 1 °Ê¾å¤Î¼«Á³¿ô¤È¤¹¤ë.(2n+1)-³Ñ·Á ¤ÈľÀþ¤Î¶¦ÍÅÀ¤¬Í¸Â¤Î»þ, ¶¦ÍÅÀ¤Î¸Ä¿ô¤ÎºÇÂç¤Ï 2n ¤Ç¤¢¤ë.ÆÌ¿³Ñ·Á¤Ë¸Â¤ë¾ì¹ç¤Ï 2 ¤Ç¤¢¤ë.
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Q ¤òÍÍý¿ôÁ´ÂΤ«¤é¤Ê¤ë½¸¹ç¤È¤·, C ¤òÊ£ÁÇ¿ôÁ´ÂΤ«¤é¤Ê¤ë½¸¹ç¤È¤¹¤ë.(a,b,c)¢º(Q-{0})¡ßQ^2 ¤ËÂФ·¤Æ y=ax^2+bx+c ¤ÎÆ󼡶ÊÀþ¤ò¹Í¤¨¤ë»þ, ĺÅÀ (-b/(2a), (-b^2+4ac)/(4a)) ¤â Q^2 ¤Î¸µ¤Ç¤¢¤ë.(a,b,c)¢º(C-{0})¡ßC^2 ¤ËÂФ·¤Æ, (-b/(2a), (-b^2+4ac)/(4a))¢º
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(3^0,3^1,3^2)=(1,3,9), (1^2,2^2,3^2)=(1,4,9).
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1^2 ¤È 12^2 ¤¬ Fibonacci ¿ô¤Ç¤¢¤ë»ö¤òµÇ°¤·¤è¤¦.
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