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Concept

Sum of squares

A number or polynomial written as a total of squares of others, as 13 is 4 plus 9. Which whole numbers are sums of two, three or four squares has exact answers, and a polynomial written this way is visibly never negative, which is how many inequalities are certified.

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

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 The fifty-four sums of four weighted squares that reach every number. The 54 universal diagonal quaternary forms, as coefficient lists: 1,1,1,1; 1,1,1,2; 1,1,1,3; 1,1,1,4; 1,1,1,5; 1,1,1,6; 1,1,1,7; 1,1,2,2; 1,1,2,3; 1,1,2,4; 1,1,2,5; 1,1,2,6; 1,1,2,7; 1,1,2,8; 1,1,2,9; 1,1,2,10; 1,1,2,11; 1,1,2,12; 1,1,2,13; 1,1,2,14; 1,1,3,3; 1,1,3,4; 1,1,3,5; 1,1,3,6; 1,2,2,2; 1,2,2,3; 1,2,2,4; 1,2,2,5; 1,2,2,6; 1,2,2,7; 1,2,3,3; 1,2,3,4; 1,2,3,5; 1,2,3,6; 1,2,3,7; 1,2,3,8; 1,2,3,9; 1,2,3,10; 1,2,4,4; 1,2,4,5; 1,2,4,6; 1,2,4,7; 1,2,4,8; 1,2,4,9; 1,2,4,10; 1,2,4,11; 1,2,4,12; 1,2,4,13; 1,2,4,14; 1,2,5,6; 1,2,5,7; 1,2,5,8; 1,2,5,9; 1,2,5,10.

Fifteen numbers decide every number

Lagrange proved that x² + y² + z² + w² takes every whole value. Ramanujan asked which other sums of four weighted squares do, and in 1917 listed fifty-five. One of them is wrong: x² + 2y² + 5z² + 5w² misses 15. The mistake points at a theorem found eighty years later — to know whether such a form reaches every number, it is enough to check that it reaches 1, 2, 3, 5, 6, 7, 10, 14 and 15. Nine checks decide infinitely many cases, and the nine numbers come out of an escalation any one can run.

geometry · Figurate numbers

Named alongside it

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

Exhaustive searchQuadratic formConjectureFigurate numbersFinite checkGnomonRiemann hypothesisTriangular numbers

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