Math 2.0

Number theory Divisibility and primes Theorem

Kummer's theorem

Builds on P-adic valuation

Statement

Let pp be a prime number, and let mm and nn be natural numbers. Then νp(m+nm)\nu_p \binom{m + n}{m} is the number of carries when one adds mm and nn in base pp. Here νp\nu_p is the p-adic valuation.

Remarks

For example, in base 22, 3+1=112+12=10023 + 1 = 11_2 + 1_2 = 100_2 has 22 carries, and (41)=4=22\binom{4}{1} = 4 = 2^2. The theorem follows from Legendre's formula in the form with the digit sums sps_p. A consequence: p∣(m+nm)p \mid \binom{m+n}{m} if and only if there is at least one carry.

History

Ernst Kummer proved the theorem in 1852.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionPrime number, definition, no evidence levelPrime numberdefinitionP-adic valuation, definition, no evidence levelP-adic valuationdefinitionKummer's theorem, statement, no evidence levelKummer's theoremstatement
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  1. Divisibility definition
  2. Prime number definition
  3. P-adic valuation definition
  4. Kummer's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Kummer%27s_theorem, Statement
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