A number theory problem by Shintaro Inaba

Number Theory Level 3

two integers m m and n n are bigger than 2 2 .

If m 3 + 1 = n 3 + 1000 m^3+1=n^3+1000 , what is m − n m-n ?


The answer is 3.

Shintaro Inaba by Shintaro Inaba , 17, Japan

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1 solution

Hana Wehbi
Aug 3, 2017

m 3 + 1 = n 3 + 1000 ⟹ m 3 − n 3 = ( m − n ) ( m 2 + m n + n 2 ) = 999 m^3+1=n^3+1000\implies m^3 - n^3 =(m-n)(m^2+mn+n^2)= 999

then we can try some values for n n and m m . The integer solutions are here m = 12 m=12 and n = 9 ⟹ m − n = 3 n=9 \implies m-n= \boxed{3}

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