Leudesdorf Theorem

Let $t(m)$ denote the set of the $\phi(m)$ numbers less than and Relatively Prime to $m$, where $\phi(n)$ is the Totient Function. Then if

$$S_m\equiv \sum_{t(m)} {1\over t},$$

then

$$\cases{ S_m\equiv 0\ \left({{\rm mod\ } {m^2}}\right) & if ... ...m mod\ } {{\textstyle{1\over 4}}m^2}}\right) & if $m=2^a$.\cr}$$

See also Bauer's Identical Congruence, Totient Function

References

Hardy, G. H. and Wright, E. M. ``A Theorem of Leudesdorf.'' §8.7 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 100-102, 1979.