No solution to phi(n)=7
\[\phi (n)=k\]
does not have a solution for all values of \[k\]
, where \[\phi (n)\]
counts the number of positive integers less than or equal to \[n\]
.\[\phi (n)= \phi (p_1^{k_1}p_2^{k_2}...p_r^{k_r}=(p_1-1)p_1^{k_1-1}(p_2-1)p_2^{k_2-1}...(p_r-1)p_r^{k_r-1}\]
If
\[\phi (n)=7 = 1 \times 7\]
then the only possible factors are \[1=(2-1), \; 7=(8-1)\]
but 8 is not a prime number, so \[\phi (n)=7\]
has no solution.