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#5
21 µØÅÒ¤Á 2006, 20:56
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¢éÍááÍФÃѺ ¼ÁËÒä´éáµè m ¡Ñº n ·ÕèÊÍ´¤Åéͧ¡Ñºà§×è͹䢢éÍ 1 ¡Ñº 2 áµè·ÓµèÍäÁèà»ç¹ÍФÃѺ
ãËéÅͧ´Ù¨Ò¡¢éÍ Êͧ ¡è͹ ¹Ð ¨Ò¡ m^2 -n^2 =2541 (m+n)(m-n)=2541 ãËé A=m+n and B=m-n áÅéÇᡵÑÇ»ÃСͺ¢Í§ 2541 ÍÍ¡ÁÒ ¨Ñºà»ç¹¤Ù³¡Ñ¹à»ç¹¤Ùèæ â´Â A>B áÅéÇá¡éÊÁ¡ÒÃËÒ¤èÒ m áÅÐ n áÅéÇàÍÒä»ËÒ ËÃÁ ¨Ðä´éÍÂÙè¤Ùèà´ÕÂÇ·Õè ËÃÁ äÁèãªè 1 ¤ÃѺ ¤×Í m áÅÐ n ÊÇè¹·ÕèàËÅ×Í·ÓäÁèà»ç¹áÅéǤÃѺ »Å.¼ÁÍÒ¨¤Ô´¼Ô´¡çä´é¹Ð
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#7
21 µØÅÒ¤Á 2006, 21:04
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àÃÕ§¢éͨҡº¹Å§ÅèÒ§¹Ð¤ÃѺ
¢éÍáá: ãËé $(m,n)=:d\ne1$ ´Ñ§¹Ñé¹ $m=pd,\ n=qd,\ p,q\in\mathbb{Z}^+$ ¨Ò¡ $m^2-n^2=d^2(p^2-q^2)=2541=11^2\cdot3\cdot7$ ¨Ð¾ºÇèÒ $d=11$ áÅÐ $p^2-q^2=21$ (·ÓäÁ) «Öè§Áդӵͺ·Õèà»ç¹¨Ó¹Ç¹àµçÁºÇ¡ÊͧªØ´¤×Í $(p,q)=(5,2),(11,10)$ àÁ×èÍ $(p,q)=(5,2)$ ¨Ðä´é $m=55,\ b=22$ ´Ñ§¹Ñé¹ $k=\frac{3003-a}{429+a}$ ËÃ×Í $a=\frac{3432}{k+1}-429$ à¾ÃÒÐ $a,k$ à»ç¹¨Ó¹Ç¹àµçÁºÇ¡ ´Ñ§¹Ñé¹ $k+1|3432$ áÅÐ $1\le{}k+1<8$ ¹Ñ蹤×Í $k=1,2,3$ àÁ×èÍ $(p,q)=(11,10)$ ¨Ðä´é $m=121,\ b=110$ ´Ñ§¹Ñé¹ $k=\frac{14531-a}{11979+a}$ ËÃ×Í $a=\frac{2552}{k+1}-11979<0$ àÁ×èÍ $k>0$ ¢éÍÊͧ: à¢Õ¹ $a_1=33b_1,\ a_2=33b_2,\ a_3=33\cdot7b_3,\ a_4=33\cdot7b_4$ ¨Ò¡à§×èÍ¹ä¢ (2) ¨Ðä´é $2b_3+b_4=11$ à¾ÃÒÐ $0<b_3<b_4$ áÅÐ $p^2\not|a_4$´Ñ§¹Ñé¹ $(b_3,b_4)=(2,7),(3,5)$ ¨Ò¡ $a_1+a_2=a_3$ àÁ×èÍ $(b_3,b_4)=(2,7)$ ¨Ðä´é $33(b_1+b_2)=14\cdot33=462$ áµèà¾ÃÒÐ $200<a_1<a_2$ ¡Ã³Õ¹Õé¨Ö§à»ç¹ä»äÁèä´é (·ÓäÁ) àÁ×èÍ $(b_3,b_4)=(3,5)$ ¨Ò¡ $(b_1,b_2)=1,\ 6<b_1<b_2$ áÅÐà§×èÍ¹ä¢ (4) ¨Ð¾ºÇèÒ $(b_1,b_2)=(8,13),\ (10,11)$ ´Ñ§¹Ñ鹤ӵͺÁÕÊͧªØ´¤×Í $(a_1,a_2,a_3,a_4)=(8\cdot33,13\cdot33,21\cdot33,35\cdot33)$ áÅÐ $(10\cdot33,11\cdot33,21\cdot33,35\cdot33)$ ¢éÍÊÒÁ: à¢Õ¹ $n=2545k+202=2002l+202$ ´Ñ§¹Ñé¹ $2545k=2002l$ áµè $(2002,2545)=1$ ´Ñ§¹Ñé¹ $k=2002,\ l=2545$ àÈɨҡ¡ÒÃËÒè֧à»ç¹ $2545\cdot2002+202\equiv15\cdot22-18\equiv37\pmod{55}$
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish. 22 µØÅÒ¤Á 2006 00:16 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ nongtum
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