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Concept

Kummer's theorem

The statement that a prime divides a binomial coefficient exactly as many times as there are carries when the two lower indices are added in that prime's base. It turns a question about an enormous number into a question about column addition, and Lucas' theorem is its case of no carries.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Pascal's triangle mod 2, 32 rows. Only the odd entries are drawn; the pattern that appears is the Sierpiński triangle.

Pascal's triangle, in two colours

Shade the odd numbers in Pascal's triangle and a fractal appears. Nothing was designed to produce it, and the same shape arrives independently from a completely different construction.

discrete · Pascals triangle 3 carries in base 2, and 2 divides it 3 times. The addition of 5 and 7 written in base 2, column by column, with the carries marked. There are 3, and 2 divides the binomial coefficient 792 exactly 3 times.

The carries decide the divisibility

How many times a prime divides a binomial coefficient is not a fact about the coefficient at all. It is a count of the carries that happen when two numbers are added in that prime's base, which is a question about column addition and has nothing to do with choosing anything.

discrete · Pascals triangle Pascal's triangle modulo 4, where one digit at a time is not enough. 32 rows of Pascal's triangle coloured by remainder modulo 4 — hue for the last base-2 digit, depth for the second. The digit-by-digit product that gives every remainder modulo 2 gets the remainder modulo 4 wrong at 100 of the 243 entries 2 does not divide.

A remainder read two digits at a time

Lucas' theorem reads a binomial coefficient's remainder on division by a prime off its digits one at a time. On division by the prime's square the same reading is wrong at four odd entries in ten. What replaces it still reads digits — in overlapping pairs, with the prime taken out first and a sign that the carries decide.

discrete · Pascals triangle

Named alongside it

The objects these essays reach for when they reach for this one.

Binomial coefficientLucas' theoremPrimesDivisibilityParityPlace valueSelf-similarityCyclic groupModular arithmeticRecursionSierpiński triangle

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