<P>장두천 선생님의 답글입니다.</P> <P> </P> <P> <OBJECT style="WIDTH: 169px; HEIGHT: 34px; TOP: 0px; LEFT: 0px" class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=169 align=absMiddle height=34><PARAM NAME="_cx" VALUE="4471"><PARAM NAME="_cy" VALUE="899"><PARAM NAME="BackColor" VALUE="4063280"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/37.8`8.2/12,[1]{{a}^{3}`plus`{b}^{3}`equal`{c}^{3}, b`equal`{a`plus`c}/{2}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT>를 만족하는 자연수 a,b,c는 존재하지 않는다.</P> <P> </P> <P>왜냐하면 주어진 조건을 만족하는 자연수 a,b,c가 존재한다면 a,b,c의 최대공약수를 G라 두면 a=Ga', b=Gb', c=Gc' 를 만족하는 자연수a',b',c'가 존재하며 a',b',c'의 최대공약수는 1 이다.</P> <P>따라서 <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=349 align=absMiddle height=30><PARAM NAME="_cx" VALUE="9233"><PARAM NAME="_cy" VALUE="793"><PARAM NAME="BackColor" VALUE="33277"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/81.3`7/12,[1]{{(Ga')}^{3}`plus`{(Gb')}^{3}`equal`{(Gc')}^{3} `arrow`{ a'}^{3}`plus`{b'}^{3}`equal`{c'}^{3}`hdot3`hdot3`{@}_{1}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT></P> <P> <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=305 align=absMiddle height=35><PARAM NAME="_cx" VALUE="8069"><PARAM NAME="_cy" VALUE="926"><PARAM NAME="BackColor" VALUE="0"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/69.3`8.2/12,[1]{Gb'`equal`{Ga'`plus`Gc'}/{2} `arrow` b'`equal`{a'`plus`c'}/{2}`hdot3`hdot3`{@}_{2}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT></P> <P> <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=23 align=absMiddle height=24><PARAM NAME="_cx" VALUE="608"><PARAM NAME="_cy" VALUE="635"><PARAM NAME="BackColor" VALUE="3932222"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/5.5`5.6/12,[1]{{@}_{2}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT>에 의하여 b'가 a', c'의 등차중항이므로 a'=b'-m, c'=b'+m <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=49 align=absMiddle height=24><PARAM NAME="_cx" VALUE="1296"><PARAM NAME="_cy" VALUE="635"><PARAM NAME="BackColor" VALUE="3932222"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/11.6`5.6/12,[1]{hdot3`hdot3`{@}_{3}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT>인 자연수 m 이 존재하며 b'와 m은 서로 소이다.(왜냐하면 a', b', c' 의 최대공약수가 1이므로) </P> <P> <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=23 align=absMiddle height=24><PARAM NAME="_cx" VALUE="608"><PARAM NAME="_cy" VALUE="635"><PARAM NAME="BackColor" VALUE="3932222"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/5.5`5.6/12,[1]{{@}_{3}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT> 을 <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=23 align=absMiddle height=24><PARAM NAME="_cx" VALUE="608"><PARAM NAME="_cy" VALUE="635"><PARAM NAME="BackColor" VALUE="3932222"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/5.5`5.6/12,[1]{{@}_{1}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT> 에 대입하여 정리하면 </P> <P> <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=289 align=absMiddle height=45><PARAM NAME="_cx" VALUE="7646"><PARAM NAME="_cy" VALUE="1190"><PARAM NAME="BackColor" VALUE="5439520"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/67`9.6/12,[1]{{b'}^{3}`equal`6{b'}^{2}m`plus`2{m}^{3} `arrow` {b'}^{2}`equal`6b'm`plus`{2{m}^{3}}/{b'}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT> 그런데 b' 와 m 이 서로 소 이므로 b'=1, 2 .</P> <P>한편 b'=1, 2 이면 위의 <OBJECT class=MATH classid=clsid:70E4BF88-2729-4A46-8794-D459BAC0EB63 width=57 align=absMiddle height=24><PARAM NAME="_cx" VALUE="1508"><PARAM NAME="_cy" VALUE="635"><PARAM NAME="BackColor" VALUE="3932222"><PARAM NAME="Enabled" VALUE="-1"><PARAM NAME="Script" VALUE="[0]#wh4/13.3`5.6/12,[1]{{@}_{1}, {@}_{2}}/{#xc333333}"><PARAM NAME="FontSize" VALUE="12"><PARAM NAME="FontColor" VALUE="3355443"><PARAM NAME="Editable" VALUE="0"><PARAM NAME="Measurable" VALUE="0"></OBJECT> 를 만족하는 자연수 a' 와 c'가 존재하지 않는다. 이 사실은 가정에 모순되므로 위의 명제는 참이다.</P>
