Let $\phi$ denote Euler's totient.
A problem from a book I red required to prove that the equation $\phi(x)+\phi(y)=\phi(x+y)$ has infinitely many solutions.
I solved it: Let's take $x=p$ and $y=2p$, where $p\ge 5$ is prime; easy to check.
Later I started looking for a solution with $\gcd(x,y)=1$, I wrote a program in C that checked many small numbers and gave thousands of examples, but I was unable to spot one particular infinite sequence among them.
How can we prove that the equation has infinitely many relatively prime solutions ?
Here is a particular result from the program I wrote, all relatively prime solutions with $1\le x<y<100$:
$(1,2), (4,5), (5,16), (6,19), (7,38), (8,17), (8,77), (11,16), (13,14), (13,20), (16,35), (24,41), (24,53), (25,26), (25,32), (26,29), (28,37), (28,65), (29,40), (29,64), (30,89), (31,86), (32,55), (36,89), (36,97), (37,50), (37,56), (38,47), (44,85), (44,89), (46,79), (48,97), (49,62), (50,91), (52,77), (53,88), (55,92), (56,89), (61,80), (62,83), (64,77), (64,95), (68,91), (76,77), (78,97).$