Selfridge-Hurwitz Residue

Let the Residue from Pépin's Theorem be

$$R_n\equiv 3^{(F_n-1)/2}\ \left({{\rm mod\ } {F_n}}\right),$$

where $F_n$ is a Fermat Number. Selfridge and Hurwitz use

$$R_n ({\rm mod\ } 2^{35}-1, 2^{36}, 2^{36}-1).$$

A nonvanishing $R_n {\rm\ (mod\ } 2^{36})$ indicates that $F_n$ is Composite for $n>5$.

See also Fermat Number, Pépin's Theorem

References

Crandall, R.; Doenias, J.; Norrie, C.; and Young, J. ``The Twenty-Second Fermat Number is Composite.'' Math. Comput. 64, 863-868, 1995.