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A positive integer m has the property that m^2 is expressible in the form 4 n^2-5 n+16 where n is an integer (of any sign). Find the maximum possible value of |m-n| .

Correct answer

14

Step-by-step solution

array ll & m^2=4 n^2-5 n+16 & 4 n^2-5 n+16-m^2=0 & 8 n=5 16 m^2-231 array Let D^2=16 m^2-231 and D>0 (4 m+D)(4 m-D)=231=7 11 13 Case-I: 4 m+D=231 and 4 m-D=1 then m=29 and D=115, n=15 |m-n|=14 Case-II: array ll & 4 m+D=77 and 4 m-D=3 & m=10, D=37 and n=-4 & |m-n|=14 array Case-III: array ll & 4 m+D=33 and 4 m-D=7 & m=5, D=13 and n=-1 & |m-n|=6 array Case-IV: array ll & 4 m+D=21 and 4 m-D=11 & m=4, D=5 and n=0 array Hence |m-n|=4 Maximum value of |m-n|=14

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