Kummer's theorem

In mathematics, Kummer's theorem for binomial coefficients gives the p-adic valuation $\nu _{p}$ of a binomial coefficient, i.e., the highest power of a prime number p dividing this binomial coefficient. The theorem is named after Ernst Kummer, who proved it in the paper Kummer (1852).

Statement

Kummer's theorem states that for given integers n ≥ m ≥ 0 and a prime number p, the p-adic valuation ${\displaystyle \nu _{p}\left({\tbinom {n}{m}}\right)}$ is equal to the number of carries when m is added to n − m in base p.

It can be proved by writing ${\tbinom {n}{m}}$ as ${\tfrac {n!}{m!(n-m)!}}$ and using Legendre's formula.

See also

• Lucas's theorem

References

• Kummer, Ernst (1852). "Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen". Journal für die reine und angewandte Mathematik. 44: 93–146. doi:10.1515/crll.1852.44.93.
• Kummer's theorem at PlanetMath.org.