Concept

Place value

A way of writing numbers in which each digit is multiplied by a power of the base according to its position, so 307 means three hundreds, no tens and seven ones. Reading digits in a prime base turns divisibility questions about binomial coefficients into questions about carries.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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 coefficient Divisibility Kummer's theorem Lucas' theorem Primes Cyclic group Modular arithmetic Parity Self-similarity

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