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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)

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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.

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Leudesdorf Theorem

Cite this as:

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

URL: https://mathworld.wolfram.com/LeudesdorfTheorem.html

⇱ Leudesdorf Theorem -- from Wolfram MathWorld

Leudesdorf Theorem

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References

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