Concept

Totient

The count of numbers below a given one sharing no factor with it. It counts the invertible elements on a dial of that size, and its product formula over the prime factors is what makes it computable.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Which regular polygons a compass and straightedge can draw, up to 100. A grid of the integers with the constructible ones filled in, each verdict computed two independent ways.

Which polygons can be drawn

Three sides yes, seven no, seventeen yes. The list of constructible regular polygons is neither everything nor almost nothing, and the pattern in it is a fact about which numbers are one less than a power of two.

computation · Constructible numbers

The non-zero elements of GF(16) as the powers of one of them. A ring of the field's non-zero elements in the order the powers of a primitive element produce them, beside a table of exponents.

Every element is a power of one of them

Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.

computation · Finite fields

The 7 7th roots of unity. 7 points spaced evenly around the unit circle, at the vertices of a regular 7-sided polygon.

The polygon an equation forces

The n solutions of z to the n equals one are the corners of a regular polygon, and nearly everything about them — that they form a group, that they sum to zero, that the equation factors into pieces with whole-number coefficients — is that picture read carefully.

algebra · Roots of unity $The fractions, put in a line. A grid whose rows are numerators and columns denominators, walked by antidiagonals, with the place each fraction takes in the list written in its cell and the repeats left blank.$

The arithmetic that loses subtraction

Adding one to an infinite collection changes nothing, and neither does doubling it, or squaring it. What that costs is the two operations that were doing the work — an equation between infinite sizes cannot be cancelled, and how many are left stops being a question.

logic · Cardinality

The coefficients of (1 + x)¹² sorted by remainder mod 3. The binomial coefficients of the 12th power coloured by the remainder of their index on division by 3, beside the 3 points one plus a root of unity, whose powers averaged pick out each colour's total.

Every third coefficient

Add every third number in the twelfth row of Pascal's triangle and the answer is 1366 — a third of 4096, rounded up. Which way the rounding goes is decided by two arrows of length one in the complex plane, and the same average over the roots of unity counts dice totals, subsets and necklaces.

algebra · Roots of unity $The 46 Farey fractions of order 12, and how far each strays from even spacing. Farey fractions of order 12 against 46 evenly spaced points, with the deviation of each drawn as a bar; largest deviation 0.0616.$

How evenly the fractions spread

List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.

number · Stern brocot

The primitive 15th roots of unity, raised to a power and added. c₁₅(1) = 1; c₁₅(3) = −2; c₁₅(5) = −4; c₁₅(6) = −2; c₁₅(10) = −4; c₁₅(15) = 8.

The sums of primitive roots are always whole

Take the roots of unity of order exactly q, raise each to the power n and add them. The answer is always a whole number, it depends on n only through what n shares with q, and as n varies the sums behave like the sines and cosines of a Fourier series — so well that Ramanujan could rebuild the sum of the divisors of any number from them, and prove that their weighted total is nought, a fact that at n = 1 is the prime number theorem.

algebra · Roots of unity

Named alongside it

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

Cyclic group Roots of unity Cyclotomic polynomial Mobius function Modular arithmetic Prime number theorem Primitive element Regular polygon Absorption Bijection Binomial coefficient Cancellation

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