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Figurate numbers

The counts of dots that fill a regular shape of growing size — triangles, squares, pentagons and on, or pyramids in three dimensions. Each is a running total of an arithmetic sequence, and which numbers are sums of a few of them has exact answers that Fermat began to state.

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

The sum of the first 6 squares, as a staircase over a curve. Bars of height k^2 for k from 1 to 6, totalling 91, drawn over the curve y = x^2, whose area up to 6 is 72.00. The slivers between staircase and curve hold 19.00, close to half the last bar.

Sums of powers, read off a staircase

Add the first n squares, or cubes, or seventh powers, and the answer is always a polynomial in n. Its first term is the area under a curve, its second is half of the last step, and every term after that is a correction for the corners of a staircase — which is where the Bernoulli numbers come from, and why they eventually grow without bound.

geometry · Figurate numbers The fewest pentagonal numbers adding to each number up to 120. Fewest pentagonal numbers summing to 1..120; the most ever needed up to 20000 is 5, by 9, 21, 31, 43, 55, 89.

Six numbers that need five pentagons

In 1638 Fermat wrote that every whole number is a sum of three triangular numbers, four squares, five pentagonal numbers, six hexagonal numbers, and so on for every polygon — and that he had a proof he would not write down. The claim is true; Cauchy proved it in 1813. What the claim hides is how unequal the cases are. Triangles and squares need their full count infinitely often. For pentagons, only six numbers ever need all five — 9, 21, 31, 43, 55 and 89 — and from hexagons on, two apiece.

geometry · Figurate numbers Numbers made of pyramids. below 1000000: 241 need five; first 17, 27, 33, 52, 73, 82, 83, 103, 107, 137, 153, 162; last 343867 = 341376+1771+680+20+20.

Five pyramids for every number

Stack balls in triangular pyramids and the piles hold 1, 4, 10, 20, 35, … balls. Frederick Pollock guessed in 1850 that every whole number is a sum of at most five such piles. Seventeen needs five — 10 + 4 + 1 + 1 + 1 — and so do 240 other numbers, the largest 343,867; past that, every number up to ten million needs four or fewer. Eight is proved to be enough. Five is not, and the reason is the same one that makes the three-dimensional cousins of the classical theorems hard.

geometry · Figurate numbers

Named alongside it

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

Exhaustive searchTriangular numbersApproximationAsymptotic seriesBinomial coefficientConjectureError termGnomonGreedy algorithmIntegralPolynomialQuadratic form

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