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ritvik1512
Peter has recently been fascinated by numbers that can be expressed in the form f(a,b,c)=a(b−c)^3+b(c−a)^3+c(a−b)^3 for some integers a,b and c. He thinks that we should use this system to do arithmetic in future. How many positive integers N≤1000 can be expressed in the form of f(a,b,c)?
f(a,b,c)=a(b−c)^3+b(c−a)^3+c(a−b)^3
Oh! Peter should try it first.
Unless it's Spider-man or Peter Griffin :D
Oh! Peter should definitely try it first.
@ritvik1512 I meant to say that, can you show us what you tried so that we can get to your mistake and problem and help accordingly.
does anybody think this might give only the even numbers?
If you manipulate the equation a bit (by expanding the cubes, and simplifying/factoring), you can show that:\[f(a,b,c)=ab(b+a)(b-a)+bc(c+b)(c-b)+ac(a-c)(a+c)\]From here is is easier to show (using a modular arithmetic argument) that no matter what integers you pick, this will always be divisible by both 2 and 3 (so divisible by 6). My guess is that you can only get the multiples of 6, and nothing else.
In fact, note that for any integer a:\[f(a,a+1,a+2)=a(-1)^3 +(a+1)(2)^3 +(a+2)(-1)^3 \]\[-a+8a+8-a-2=6a+6=6(a+1)\]So you can get any multiple of 6 you want.