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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=sum_(t(m))1/t,
(1)

then

{S_m=0 (mod m^2) if 2m, 3m; S_m=0 (mod 1/3m^2) if 2m, 3|m; S_m=0 (mod 1/2m^2) 2|m, 3m, m not a power of 2; S_m=0 (mod 1/6m^2) if 2|m, 3|m; S_m=0 (mod 1/4m^2) if m=2^a.
(2)

See also

Bauer's Identical Congruence, Totient Function

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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.

Referenced on Wolfram|Alpha

Leudesdorf Theorem

Cite this as:

Weisstein, Eric W. "Leudesdorf Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeudesdorfTheorem.html

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