# Can we find all solutions to the equation $\frac{\phi(n)}{\phi(n-1)}=5$, where $\phi(n)$ denotes the totient function?

The solutions of the equation $$\frac{\phi(n)}{\phi(n-1)}=5$$ upto $n=10^8$ , where $\phi(n)$ denotes the totient function, are :

? for(n=2,10^8,if(eulerphi(n)/eulerphi(n-1)==5,print(n,"  ",factor(n-1),"   ",fa
ctor(n))))
11242771  [2, 1; 3, 1; 5, 1; 7, 1; 11, 1; 31, 1; 157, 1]   [1171, 1; 9601, 1]
18673201  [2, 4; 3, 3; 5, 2; 7, 1; 13, 1; 19, 1]   [2161, 1; 8641, 1]
77805001  [2, 3; 3, 2; 5, 4; 7, 1; 13, 1; 19, 1]   [1801, 1; 43201, 1]
?


In the two last solutions, the two $n-1$-numbers share the same prime factors, but the first solution is completely different with a squarefree $n-1$-number.

• Can we somehow classify all the solutions of this equation ?

• What, if we replace $5$ by another positive integer ?

• These numbers are tabulated at oeis.org/A201253. The 92nd such number is 996629647861. You might try factoring them and looking for patterns. There are also tabulations at oeis for 5 replaced by a smaller positive integer. – Gerry Myerson May 18 '18 at 7:20