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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

$$n(n-2016)=m^2$$$$n^2-2016n-m^2=0$$$$n=1008+\sqrt{{1008}^2+m^2}$$$${1008}^2+m^2=u^2$$$$(u-m)(u+m)={1008}^2=2^8{3}^4{7}^2=a\times b$$$$aandbareevenfactorsof{1008}^2.$$$$u-m=a$$$$u+m=b$$$$u=\frac{a+b}{2}$$$$m=\frac{b-a}{2}$$$$n=1008+u$$$$a/b=2/508032\Rightarrow u=254017\Rightarrow n=255025$$$$a/b=4/254016\Rightarrow u=127010\Rightarrow n=128018$$$$a/b=8/127008\Rightarrow u=63508\Rightarrow n=64516$$$$a/b=16/63504\Rightarrow u=31760\Rightarrow n=32768$$$$a/b=32/31752\Rightarrow u=15892\Rightarrow n=16900$$$$a/b=64/15876\Rightarrow u=7970\Rightarrow n=8978$$$$a/b=128/7938\Rightarrow u=4033\Rightarrow n=5041$$$$a/b=6/169344\Rightarrow u=84675\Rightarrow n=85683$$$$a/b=12/.....\Rightarrow n=43350$$$$a/b=24/.....\Rightarrow n=22188$$$$a/b=48/.....\Rightarrow n=11616$$$$a/b=96/.....\Rightarrow n=6348$$$$a/b=192/.....\Rightarrow n=3750$$$$a/b=384/.....\Rightarrow n=2523$$$$a/b=3^2\times 2/.....\Rightarrow n=29241$$$$a/b=3^2\times 2^2/.....\Rightarrow n=15138$$$$a/b=3^2\times 2^3/.....\Rightarrow n=8100$$$$a/b=3^2\times 2^4/.....\Rightarrow n=4608$$$$a/b=3^2\times 2^5/.....\Rightarrow n=2916$$$$a/b=3^2\times 2^6/.....\Rightarrow n=2178$$$$a/b=3^2\times 2^7/.....\Rightarrow n=2025$$$$a/b=3^3\times 2/.....\Rightarrow n=10443$$$$a/b=3^3\times 2^2/.....\Rightarrow n=5766$$$$a/b=3^3\times 2^3/.....\Rightarrow n=3468$$$$a/b=3^3\times 2^4/.....\Rightarrow n=2400$$$$a/b=3^3\times 2^5/.....\Rightarrow n=2028$$$$a/b=3^3\times 2^6/.....\Rightarrow n=2166$$$$a/b=3^3\times 2^7/.....\Rightarrow n=2883$$$$a/b=3^4\times 2/.....\Rightarrow n=4225$$$$a/b=3^4\times 2^2/.....\Rightarrow n=2738$$$$a/b=3^4\times 2^3/.....\Rightarrow n=2116$$$$a/b=3^4\times 2^4/.....\Rightarrow n=2048$$$$a/b=3^4\times 2^5/.....\Rightarrow n=2500$$$$a/b=3^4\times 2^6/.....\Rightarrow n=3698$$$$a/b=3^4\times 2^7/.....\Rightarrow n=6241$$$$a/b=7\times 2/.....\Rightarrow n=37303$$$$a/b=7\times 2^2/.....\Rightarrow n=19166$$$$a/b=7\times 2^3/.....\Rightarrow n=10108$$$$a/b=7\times 2^4/.....\Rightarrow n=5600$$$$a/b=7\times 2^5/.....\Rightarrow n=3388$$$$a/b=7\times 2^6/.....\Rightarrow n=2366$$$$a/b=7\times 2^7/.....\Rightarrow n=2023$$$$a/b=7^2\times 2/.....\Rightarrow n=6241$$$$a/b=7^2\times 2^2/.....\Rightarrow n=3698$$$$a/b=7^2\times 2^3/.....\Rightarrow n=2500$$$$a/b=7^2\times 2^4/.....\Rightarrow n=2048$$$$a/b=7^2\times 2^5/.....\Rightarrow n=2116$$$$a/b=7^2\times 2^6/.....\Rightarrow n=2738$$$$a/b=7^2\times 2^7/.....\Rightarrow n=4225$$$$.....etc.$$$$generallya=3^i{7}^j{2}^{k}andb=3^{4-i}{7}^{2-j}{2}^{8-k}$$$$with0⩽i⩽4,0⩽j⩽2,1⩽k⩽7anda⩽b.$$$$since5\times 3\times 7=105and\left[105/2\right]+1=53,$$$$therearetotally53possiblevalues$$$$fornsuchthatn(n-2016)is$$$$aperfectsquare.$$

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