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 h(a,b,c) as the expression in f. Let r,s,t be three integers > 1 such that a + b + c = r + s + t. Then h(a,b,c) = h(r,s,t). In particular, we have
h(a,b,c) = h(2,2,z-4)
= 2f^{4z- 18}(4z -16) + f^{3z -12}(4z -16) = z
where I have introduced z = a + b + c
For instance, if we take a=b=c=2 we get
f^6(8) = 2
Edited on August 5, 2021, 11:38 am
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Posted by FrankM
on 2021-08-04 20:28:44 |
Further thoughts (2)
