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Date: 10-5-2020
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Let denote the set of the numbers less than and relatively prime to , where is the totient function. Define
(1) |
Then a theorem of Lagrange states that
(2) |
for an odd prime (Hardy and Wright 1979, p. 98). Actually, this relationship holds for some composite values as well. Value for which it holds are , 3, 4, 5, 6, 7, 10, 11, 13, 17, 19, 23, 29, ... (OEIS A158008).
This can be generalized as follows. Let be an odd prime divisor of and the highest power which divides , then
(3) |
and, in particular,
(4) |
Now, if is even and is the highest power of 2 that divides , then
(5) |
and, in particular,
(6) |
REFERENCES:
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.
Sloane, N. J. A. Sequence A158008 in "The On-Line Encyclopedia of Integer Sequences."
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