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Question Number 130085 by liberty last updated on 22/Jan/21
Show that if { ((a=r^2 −2rs−s^2 )),((b=r^2 +s^2 )),((c=r^2 +2rs−s^2 )) :} for some integers r,s then a^2 ,b^2 ,c^2 are three square in AP.
Showthatif{a=r2−2rs−s2b=r2+s2c=r2+2rs−s2forsomeintegersr,sthena2,b2,c2arethreesquareinAP.
Answered by benjo_mathlover last updated on 22/Jan/21
a^2 ,b^2 ,c^2 in AP then 2b^2 = a^2 +c^2 (1)2b^2 =2(r^2 +s^2 )^2 =2(r^4 +2r^2 s^2 +s^4 ) = 2r^4 +4r^2 s^2 +2s^4 (2) a^2 +c^2 =(a+c)^2 −2ac =(2r^2 −2s^2 )^2 −2(r^2 −s^2 −2rs)(r^2 −s^2 +2rs) =4r^4 −8r^2 s^2 +4s^4 −2[ (r^2 −s^2 )^2 −4r^2 s^2 ] = 4r^4 −8r^2 s^2 +4s^4 −2(r^4 −6r^2 s^2 +s^4 ) = 2r^4 +4r^2 s^2 +2s^2 It follows that a^2 ,b^2 ,c^2 in AP
a2,b2,c2inAPthen2b2=a2+c2(1)2b2=2(r2+s2)2=2(r4+2r2s2+s4)=2r4+4r2s2+2s4(2)a2+c2=(a+c)2−2ac=(2r2−2s2)2−2(r2−s2−2rs)(r2−s2+2rs)=4r4−8r2s2+4s4−2[(r2−s2)2−4r2s2]=4r4−8r2s2+4s4−2(r4−6r2s2+s4)=2r4+4r2s2+2s2Itfollowsthata2,b2,c2inAP

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