tangpakchiu 寫到:#ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl#設p,q,r為質數,試求滿足條件p^3=p^2+q^2+r^2之所有可能值 #ed_op#/font#ed_cl##ed_op#/span#ed_cl##ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl##ed_op#/font#ed_cl##ed_op#/span#ed_cl# #ed_op#/div#ed_cl##ed_op#div style="color: rgb(255, 0, 0);"#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl#Obviously,p is greater than or equal to q and r.#ed_op#/font#ed_cl##ed_op#/span#ed_cl##ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl#WLOG, letp>=q>=r#ed_op#/font#ed_cl##ed_op#/span#ed_cl##ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl##ed_op#/font#ed_cl##ed_op#/span#ed_cl# #ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl#p^3=p^2+q^2+r^2=<3p^2#ed_op#/font#ed_cl##ed_op#/span#ed_cl##ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl##ed_op#/font#ed_cl##ed_op#/span#ed_cl# #ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl#p=<3#ed_op#/font#ed_cl##ed_op#/span#ed_cl##ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl##ed_op#/font#ed_cl##ed_op#/span#ed_cl# #ed_op#/div#ed_cl##ed_op#div#ed_cl##ed_op#span class="postbody"#ed_cl##ed_op#font size="2"#ed_cl#therefore, there is only one solution,(3,3,3)#ed_op#/font#ed_cl##ed_op#/span#ed_cl##ed_op#/div#ed_cl#
#ed_op#br#ed_cl#為何不可以 p<q 或 p<r ??#ed_op#br#ed_cl#e.g. p=11, q=11 (只是舉正整數例子,並非質數例子)#ed_op#br#ed_cl#p#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#(p-1)=q#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#+r#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl# => 11#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#(10)-11#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#=r#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl# => 1089=r#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl# => r=33#ed_op#br#ed_cl#=> r > p !!!#ed_op#br#ed_cl##ed_op#br#ed_cl##ed_op#br#ed_cl#該是 p is obviously less than or equal to q or r!!#ed_op#br#ed_cl#if p > q and p>r,#ed_op#br#ed_cl# p#ed_op#sup#ed_cl#3#ed_op#/sup#ed_cl#=p#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#+q#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#+r#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#<3p#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl# => p#ed_op#sup#ed_cl#2#ed_op#/sup#ed_cl#(p-3)<0 => p<0 -- contradiction!! => p>=q or p>=r !!!#ed_op#br#ed_cl##ed_op#br#ed_cl#雖然(3,3,3)是暫時找到的質數的唯一解...但未能證明其唯一性!...#ed_op#br#ed_cl#