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The smallest exponent $e$ for which $b^e\equiv 1\ \left({{\rm mod\ } {p}}\right)$, where $b$ and $p$ are given numbers, is the haupt-exponent of $b$ (mod $p$). The number of bases having a haupt-exponent $e$ is $\phi(e)$, where $\phi(e)$ is the Totient Function. Cunningham (1922) published the haupt-exponents for primes to 25409 and bases 2, 3, 5, 6, 7, 10, 11, and 12.

See also Complete Residue System, Residue Index


Cunningham, A. Haupt-Exponents, Residue Indices, Primitive Roots. London: F. Hodgson, 1922.

© 1996-9 Eric W. Weisstein