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Concept

Hensel lemma

The rule that a root of a polynomial modulo a prime, at which the derivative is not divisible by that prime, extends to a root modulo every power of it. Each step solves a linear congruence for one more digit, which is Newton's method carried out in p-adic arithmetic.

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

Square roots of 2, lifted from one power of 7 to the next. A tree whose rows are the square roots of 2 modulo 7 to the powers 1 to 4, each joined to the root it reduces to; counts 2, 2, 2, 2.

A root lifted one digit at a time

On a dial of seven, 3 × 3 is 2. On a dial of forty-nine the square root of 2 must reduce to 3, so there are only seven candidates, and exactly one of them works: 10. On a dial of 343 exactly one lift of 10 works: 108. Each step adds one digit on the left, found by solving a linear equation, and the digits go on for ever — a number …21216213 whose square is 2, in a world where closeness means divisibility by seven.

discrete · Modular arithmetic Powers of 5 and 6 until the digits settle. five-adic tail 0896109004106619977392256259918212890625; six tail 9103890995893380022607743740081787109376; settled after 8: 10, 9.

A number that is its own square

Square 625 and the answer, 390625, ends in 625. Square 9376 and it ends in 9376. Keep extending them and two endless strings of digits appear, running leftward, each equal to its own square — and their product is nought, though neither is. Base ten does this because ten is two times five: a number in base ten is secretly a pair of numbers, one in base two and one in base five, and these two strings are the pair (one, nought) and (nought, one). In a prime base no such thing can happen.

discrete · Modular arithmetic

Named alongside it

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

LiftingModular arithmeticNewtons methodP adic numbersChinese remainder theoremDerivativeLocal global principleQuadratic residueSquare rootZero divisor

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