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)
Define arguments of interest as the product of at least three primes. Note that for ay integer, k, that is not of interest, f(k) is free to take on any value without affecting the validity of the specifying equation.
Let K = abc be an argument of interest and assume that f(abc) = abc. Then abc = a + b + c, which is not true for integers a,b,c > 1. Therefore f has no fixed points among arguments of interest.
|
|
Posted by FrankM
on 2021-08-16 22:06:16 |
No fixed point for arguments of interest
