Given $n$ find all values n such that: $\phi(n) = 26$.

I've searched over the web and I've managed to find the lower and upper bounds for n, but i don't know how to go on from this point.

I'll be glad for some directions.



Suppose $\phi(n)=26$ where $\displaystyle n=p_1^{a_1}\cdots p_r^{a_r}$.

Then $\displaystyle p_1^{a_1-1}\cdots p_r^{a_r-1}(p_1-1)\cdots(p_r-1)=26$, so $13$ must divide one of the factors on the left side.

If $13=p_i$ for some $i$, we get a contradiction since then $12\lvert26$.

If $13\lvert p_i-1$ for some $i$, we get a contradiction since then $p_i=14$ or $p_i=27$.

Therefore there are no solutions to this equation.


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