Concept

Totient

The count of numbers below a given one sharing no factor with it.

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

3 · 4 · 5 · 6 · 8 · 10 · 12 · 15 · 16 · 17 · 20 · 24 · …3φ24φ25φ46φ27φ68φ49φ610φ411φ1012φ413φ1214φ615φ816φ817φ1618φ619φ1820φ821φ1222φ1023φ2224φ825φ2026φ1227φ1828φ1229φ2830φ831φ3032φ1633φ2034φ1635φ2436φ1237φ3638φ1839φ2440φ1641φ4042φ1243φ4244φ2045φ2446φ2247φ4648φ1649φ4250φ2051φ3252φ2453φ5254φ1855φ4056φ2457φ3658φ2859φ5860φ1661φ6062φ3063φ3664φ3265φ4866φ2067φ6668φ3269φ4470φ2471φ7072φ2473φ7274φ3675φ4076φ3677φ6078φ2479φ7880φ3281φ5482φ4083φ8284φ2485φ6486φ4287φ5688φ4089φ8890φ2491φ7292φ4493φ6094φ4695φ7296φ3297φ9698φ4299φ60100φ40n = 3 to 100: 24 constructible, 74 notdecided twice — by the Fermat-prime criterion and by φ(n) being a power of two — and thetwo agreed at every one of the 98

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
1αα²α³α+1α²+αα³+α²α³+α+1α²+1α³+αα²+α+1α³+α²+αα³+α²+α+1α³+α²+1α³+1GF(16)*15 elementspowerelementα⁰1α¹αα²α²α³α³α⁴α+1α⁵α²+αα⁶α³+α²α⁷α³+α+1α⁸α²+1α⁹α³+αα¹⁰α²+α+1α¹¹α³+α²+αα¹²α³+α²+α+1α¹³α³+α²+1α¹⁴α³+1every non-zero element of GF(16) is a power of α — the 15 of them, once each, around the wheel8 of the 15 elements generate the whole group, which is φ(15) = 8, and the product rule waschecked on all 225 pairs

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

Named alongside it

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

Cyclic groupConstructible numberCounting argumentDegree of an extensionDiscrete logarithmFermat primeFermats little theoremFinite fieldModular arithmeticPrimitive elementRational root theoremRegular polygon

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