As of late I have been researching Euler's Totient function. For the last week or so I have specifically been studying the equation: $\varphi(a)+\varphi(b)=\varphi(a+b)$
While the equation isn't true for all $a$ and $b$ I believe I've found the criteria that need to be met for the equation to be true.
I have tried to find information regarding the equation but haven't had much luck. I would like to know what research has been done previously on this equation to see if my results are, by any chance, new.
Thanks for the help!
Here is my theorem:
For $a≠b$, $φ(a)+φ(b)=φ(a+b)$ when $a=2^m p_1 p_2…p_k$ and $b=2^n p_1 p_2…p_k p_{k+1}$.
In addition to the extra requirements that $m$ and $n$ be $≥0$ but both may not be $0$, (in other words, either $a$ or $b$ may be odd but they cannot both be odd) and $2^{m-n}+p_{k+1}$ must be prime.
In addition I have used a slightly different representation for $a$ and $b$ than the standard canonical representation, but the $p$ terms do not all have to be distinct.
Ex: $a=2^3(3)(3)$ and $b=2^2(3)(3)(11)$
The equation is also true if $a=b$ and $a,b$ are even.