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Describes the highest power of primes dividing a binomial coefficient From Wikipedia, the free encyclopedia
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).
Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation of the binomial coefficient 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]
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 3.
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
Kummer's theorem can be generalized to multinomial coefficients as follows:
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