The solutions of the equation $$|\varphi(n+1)-\varphi(n)|=2$$ upto $\ \ n=10^8-1\ \ $ are (the first entries of the arrays) :
? j=1;a=[1,1];while(j<10^8,j=j+1;[a,b]=[b,eulerphi(j)];if(abs(a-b)==2,print([j-1
,j,a,b])))
[4, 5, 2, 4]
[5, 6, 4, 2]
[7, 8, 6, 4]
[8, 9, 4, 6]
[9, 10, 6, 4]
[14, 15, 6, 8]
[21, 22, 12, 10]
[45, 46, 24, 22]
[165, 166, 80, 82]
[254, 255, 126, 128]
[765, 766, 384, 382]
[64005, 64006, 32000, 32002]
[41067645, 41067646, 20533824, 20533822]
?
Are there infinite many solutions of this equation ?
I noticed that the even number of the pair $(n,n+1)$ has the form $2p$ with $p$ odd prime for $n\ge 9$. Is this actually a necessary condition for $n\ge 9$ and does it help to decide whether infinite or finite many solutions exist ?