Let N be the set of positive integers. Find all functions f:N->N that satisfy the equation
fabc-a(abc) + fabc-b(abc) + fabc-c(abc) = a + b + c
for all a, b, c ≥ 2.
(Here f1(n) = f(n) and fk(n) = f(fk-1(n)) for every integer k greater than 1)
(Also note: abc is the product a·b·c and not the concatenation 100a+10b+c)
Consider the sequence f(x^3), f^2(x^3), f^3(x^3), ... for some cube x^3. Either this sequence repeats after some number of steps, or else each of the terms is distinct.
Suppose the sequence repeated after x terms. Then for any natural number m, we would have f^{mx}(x^3) = f(x^3). Taking a = b = c = x, then each of the terms on the left hand side of the functional equation is equal to f(x^3), while each of the terms on the right hand side equals x.
Hmm.
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Posted by FrankM
on 2021-08-05 11:58:15 |
Further thoughts (3)
