Let T(m) denote the set of the numbers less than and relatively prime to m, where
is the
totient function. Define
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(1) |
Then a theorem of Lagrange states that
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(2) |
for p an odd prime (Hardy and Wright 1979, p. 98).
This can be generalized as follows. Let p be an odd prime divisor of m and the highest
power which divides m, then
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(3) |
and, in particular,
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(4) |
Now, if m > 2 is even and is the highest power of 2 that divides m, then
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(5) |
and, in particular,
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(6) |
Congruence, Leudesdorf Theorem
Bauer. Nouvelles annales 2, 256-264, 1902.
Hardy, G. H. and Wright, E. M. J. London Math. Soc. 9, 38-41 and 240, 1934.
Hardy, G. H. and Wright, E. M. "Bauer's Identical Congruence." §8.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 98-100, 1979.

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