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Question Number 55410 by naka3546 last updated on 23/Feb/19

Answered by mr W last updated on 24/Feb/19

n(n−2016)=m^2   n^2 −2016n−m^2 =0  n=1008+(√(1008^2 +m^2 ))  1008^2 +m^2 =u^2   (u−m)(u+m)=1008^2 =2^8 3^4 7^2 =a×b  a and b are even factors of 1008^2 .  u−m=a  u+m=b  u=((a+b)/2)  m=((b−a)/2)  n=1008+u  a/b=2/508032⇒u=254017⇒n=255025  a/b=4/254016⇒u=127010⇒n=128018  a/b=8/127008⇒u=63508⇒n=64516  a/b=16/63504⇒u=31760⇒n=32768  a/b=32/31752⇒u=15892⇒n=16900  a/b=64/15876⇒u=7970⇒n=8978  a/b=128/7938⇒u=4033⇒n=5041  a/b=6/169344⇒u=84675⇒n=85683  a/b=12/.....⇒n=43350  a/b=24/.....⇒n=22188  a/b=48/.....⇒n=11616  a/b=96/.....⇒n=6348  a/b=192/.....⇒n=3750  a/b=384/.....⇒n=2523  a/b=3^2 ×2/.....⇒n=29241  a/b=3^2 ×2^2 /.....⇒n=15138  a/b=3^2 ×2^3 /.....⇒n=8100  a/b=3^2 ×2^4 /.....⇒n=4608  a/b=3^2 ×2^5 /.....⇒n=2916  a/b=3^2 ×2^6 /.....⇒n=2178  a/b=3^2 ×2^7 /.....⇒n=2025  a/b=3^3 ×2/.....⇒n=10443  a/b=3^3 ×2^2 /.....⇒n=5766  a/b=3^3 ×2^3 /.....⇒n=3468  a/b=3^3 ×2^4 /.....⇒n=2400  a/b=3^3 ×2^5 /.....⇒n=2028  a/b=3^3 ×2^6 /.....⇒n=2166  a/b=3^3 ×2^7 /.....⇒n=2883  a/b=3^4 ×2/.....⇒n=4225  a/b=3^4 ×2^2 /.....⇒n=2738  a/b=3^4 ×2^3 /.....⇒n=2116  a/b=3^4 ×2^4 /.....⇒n=2048  a/b=3^4 ×2^5 /.....⇒n=2500  a/b=3^4 ×2^6 /.....⇒n=3698  a/b=3^4 ×2^7 /.....⇒n=6241  a/b=7×2/.....⇒n=37303  a/b=7×2^2 /.....⇒n=19166  a/b=7×2^3 /.....⇒n=10108  a/b=7×2^4 /.....⇒n=5600  a/b=7×2^5 /.....⇒n=3388  a/b=7×2^6 /.....⇒n=2366  a/b=7×2^7 /.....⇒n=2023  a/b=7^2 ×2/.....⇒n=6241  a/b=7^2 ×2^2 /.....⇒n=3698  a/b=7^2 ×2^3 /.....⇒n=2500  a/b=7^2 ×2^4 /.....⇒n=2048  a/b=7^2 ×2^5 /.....⇒n=2116  a/b=7^2 ×2^6 /.....⇒n=2738  a/b=7^2 ×2^7 /.....⇒n=4225  ..... etc.  generally a=3^i 7^j 2^k  and b=3^(4−i) 7^(2−j) 2^(8−k)   with 0≤i≤4, 0≤j≤2, 1≤k≤7 and a≤b.  since 5×3×7=105 and [105/2]+1=53,  there are totally 53 possible values  for n such that n(n−2016) is  a perfect square.

$${n}\left({n}−\mathrm{2016}\right)={m}^{\mathrm{2}} \\ $$$${n}^{\mathrm{2}} −\mathrm{2016}{n}−{m}^{\mathrm{2}} =\mathrm{0} \\ $$$${n}=\mathrm{1008}+\sqrt{\mathrm{1008}^{\mathrm{2}} +{m}^{\mathrm{2}} } \\ $$$$\mathrm{1008}^{\mathrm{2}} +{m}^{\mathrm{2}} ={u}^{\mathrm{2}} \\ $$$$\left({u}−{m}\right)\left({u}+{m}\right)=\mathrm{1008}^{\mathrm{2}} =\mathrm{2}^{\mathrm{8}} \mathrm{3}^{\mathrm{4}} \mathrm{7}^{\mathrm{2}} ={a}×{b} \\ $$$${a}\:{and}\:{b}\:{are}\:{even}\:{factors}\:{of}\:\mathrm{1008}^{\mathrm{2}} . \\ $$$${u}−{m}={a} \\ $$$${u}+{m}={b} \\ $$$${u}=\frac{{a}+{b}}{\mathrm{2}} \\ $$$${m}=\frac{{b}−{a}}{\mathrm{2}} \\ $$$${n}=\mathrm{1008}+{u} \\ $$$${a}/{b}=\mathrm{2}/\mathrm{508032}\Rightarrow{u}=\mathrm{254017}\Rightarrow{n}=\mathrm{255025} \\ $$$${a}/{b}=\mathrm{4}/\mathrm{254016}\Rightarrow{u}=\mathrm{127010}\Rightarrow{n}=\mathrm{128018} \\ $$$${a}/{b}=\mathrm{8}/\mathrm{127008}\Rightarrow{u}=\mathrm{63508}\Rightarrow{n}=\mathrm{64516} \\ $$$${a}/{b}=\mathrm{16}/\mathrm{63504}\Rightarrow{u}=\mathrm{31760}\Rightarrow{n}=\mathrm{32768} \\ $$$${a}/{b}=\mathrm{32}/\mathrm{31752}\Rightarrow{u}=\mathrm{15892}\Rightarrow{n}=\mathrm{16900} \\ $$$${a}/{b}=\mathrm{64}/\mathrm{15876}\Rightarrow{u}=\mathrm{7970}\Rightarrow{n}=\mathrm{8978} \\ $$$${a}/{b}=\mathrm{128}/\mathrm{7938}\Rightarrow{u}=\mathrm{4033}\Rightarrow{n}=\mathrm{5041} \\ $$$${a}/{b}=\mathrm{6}/\mathrm{169344}\Rightarrow{u}=\mathrm{84675}\Rightarrow{n}=\mathrm{85683} \\ $$$${a}/{b}=\mathrm{12}/.....\Rightarrow{n}=\mathrm{43350} \\ $$$${a}/{b}=\mathrm{24}/.....\Rightarrow{n}=\mathrm{22188} \\ $$$${a}/{b}=\mathrm{48}/.....\Rightarrow{n}=\mathrm{11616} \\ $$$${a}/{b}=\mathrm{96}/.....\Rightarrow{n}=\mathrm{6348} \\ $$$${a}/{b}=\mathrm{192}/.....\Rightarrow{n}=\mathrm{3750} \\ $$$${a}/{b}=\mathrm{384}/.....\Rightarrow{n}=\mathrm{2523} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{2}} ×\mathrm{2}/.....\Rightarrow{n}=\mathrm{29241} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{2}} /.....\Rightarrow{n}=\mathrm{15138} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{3}} /.....\Rightarrow{n}=\mathrm{8100} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{4}} /.....\Rightarrow{n}=\mathrm{4608} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{5}} /.....\Rightarrow{n}=\mathrm{2916} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{6}} /.....\Rightarrow{n}=\mathrm{2178} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{7}} /.....\Rightarrow{n}=\mathrm{2025} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{3}} ×\mathrm{2}/.....\Rightarrow{n}=\mathrm{10443} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{3}} ×\mathrm{2}^{\mathrm{2}} /.....\Rightarrow{n}=\mathrm{5766} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{3}} ×\mathrm{2}^{\mathrm{3}} /.....\Rightarrow{n}=\mathrm{3468} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{3}} ×\mathrm{2}^{\mathrm{4}} /.....\Rightarrow{n}=\mathrm{2400} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{3}} ×\mathrm{2}^{\mathrm{5}} /.....\Rightarrow{n}=\mathrm{2028} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{3}} ×\mathrm{2}^{\mathrm{6}} /.....\Rightarrow{n}=\mathrm{2166} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{3}} ×\mathrm{2}^{\mathrm{7}} /.....\Rightarrow{n}=\mathrm{2883} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{4}} ×\mathrm{2}/.....\Rightarrow{n}=\mathrm{4225} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{4}} ×\mathrm{2}^{\mathrm{2}} /.....\Rightarrow{n}=\mathrm{2738} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{4}} ×\mathrm{2}^{\mathrm{3}} /.....\Rightarrow{n}=\mathrm{2116} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{4}} ×\mathrm{2}^{\mathrm{4}} /.....\Rightarrow{n}=\mathrm{2048} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{4}} ×\mathrm{2}^{\mathrm{5}} /.....\Rightarrow{n}=\mathrm{2500} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{4}} ×\mathrm{2}^{\mathrm{6}} /.....\Rightarrow{n}=\mathrm{3698} \\ $$$${a}/{b}=\mathrm{3}^{\mathrm{4}} ×\mathrm{2}^{\mathrm{7}} /.....\Rightarrow{n}=\mathrm{6241} \\ $$$${a}/{b}=\mathrm{7}×\mathrm{2}/.....\Rightarrow{n}=\mathrm{37303} \\ $$$${a}/{b}=\mathrm{7}×\mathrm{2}^{\mathrm{2}} /.....\Rightarrow{n}=\mathrm{19166} \\ $$$${a}/{b}=\mathrm{7}×\mathrm{2}^{\mathrm{3}} /.....\Rightarrow{n}=\mathrm{10108} \\ $$$${a}/{b}=\mathrm{7}×\mathrm{2}^{\mathrm{4}} /.....\Rightarrow{n}=\mathrm{5600} \\ $$$${a}/{b}=\mathrm{7}×\mathrm{2}^{\mathrm{5}} /.....\Rightarrow{n}=\mathrm{3388} \\ $$$${a}/{b}=\mathrm{7}×\mathrm{2}^{\mathrm{6}} /.....\Rightarrow{n}=\mathrm{2366} \\ $$$${a}/{b}=\mathrm{7}×\mathrm{2}^{\mathrm{7}} /.....\Rightarrow{n}=\mathrm{2023} \\ $$$${a}/{b}=\mathrm{7}^{\mathrm{2}} ×\mathrm{2}/.....\Rightarrow{n}=\mathrm{6241} \\ $$$${a}/{b}=\mathrm{7}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{2}} /.....\Rightarrow{n}=\mathrm{3698} \\ $$$${a}/{b}=\mathrm{7}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{3}} /.....\Rightarrow{n}=\mathrm{2500} \\ $$$${a}/{b}=\mathrm{7}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{4}} /.....\Rightarrow{n}=\mathrm{2048} \\ $$$${a}/{b}=\mathrm{7}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{5}} /.....\Rightarrow{n}=\mathrm{2116} \\ $$$${a}/{b}=\mathrm{7}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{6}} /.....\Rightarrow{n}=\mathrm{2738} \\ $$$${a}/{b}=\mathrm{7}^{\mathrm{2}} ×\mathrm{2}^{\mathrm{7}} /.....\Rightarrow{n}=\mathrm{4225} \\ $$$$.....\:{etc}. \\ $$$${generally}\:{a}=\mathrm{3}^{{i}} \mathrm{7}^{{j}} \mathrm{2}^{{k}} \:{and}\:{b}=\mathrm{3}^{\mathrm{4}−{i}} \mathrm{7}^{\mathrm{2}−{j}} \mathrm{2}^{\mathrm{8}−{k}} \\ $$$${with}\:\mathrm{0}\leqslant{i}\leqslant\mathrm{4},\:\mathrm{0}\leqslant{j}\leqslant\mathrm{2},\:\mathrm{1}\leqslant{k}\leqslant\mathrm{7}\:{and}\:{a}\leqslant{b}. \\ $$$${since}\:\mathrm{5}×\mathrm{3}×\mathrm{7}=\mathrm{105}\:{and}\:\left[\mathrm{105}/\mathrm{2}\right]+\mathrm{1}=\mathrm{53}, \\ $$$${there}\:{are}\:{totally}\:\mathrm{53}\:{possible}\:{values} \\ $$$${for}\:{n}\:{such}\:{that}\:{n}\left({n}−\mathrm{2016}\right)\:{is} \\ $$$${a}\:{perfect}\:{square}. \\ $$

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