Kummer's theorem
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The theorem is named after Ernst Kummer, who proved it in a paper, (Kummer 1852).
Statement
Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation is equal to the number of carries when m is added to n − m in base p.
An equivalent formation of the theorem is as follows:
Write the base- expansion of the integer as , and define to be the sum of the base- digits. Then
The theorem can be proved by writing as and using Legendre's formula.[1]
Examples
To compute the largest power of 2 dividing the binomial coefficient write m = 3 and n − m = 7 in base p = 2 as 3 = 112 and 7 = 1112. Carrying out the addition 112 + 1112 = 10102 in base 2 requires three carries:
1 1 1 1 1 2 + 1 1 1 2 1 0 1 0 2
Therefore the largest power of 2 that divides is 23.
Alternatively, the form involving sums of digits can be used. The sums of digits of 3, 7, and 10 in base 2 are , , and respectively. Then
Multinomial coefficient generalization
Kummer's theorem can be generalized to multinomial coefficients as follows:
See also
References
- ^ Mihet, Dorel (December 2010). "Legendre's and Kummer's Theorems Again". Resonance. 15 (12): 1111–1121.
- Kummer, Ernst (1852). "Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen". Journal für die reine und angewandte Mathematik. 1852 (44): 93–146. doi:10.1515/crll.1852.44.93.
- Kummer's theorem at PlanetMath.