THE FOLD / BOSS / THE CHOKE POINT / THE THREE SQUARES
THE THREE SQUARES
which numbers are sums of three squares
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Legendre’s three-square theorem pins down exactly which numbers are a sum of three squares. A non-negative integer n is expressible as a2 + b2 + c2 if and only if n is not of the form 4a(8b + 7). So 7 fails (it is 8·0+7), 28 = 4·7 fails, 15 fails — but every other number, from 6 = 1+1+4 to 30 = 1+4+25, works. It is the sharp companion to Lagrange’s four-square theorem (four squares always suffice): three squares suffice for all but a thin, precisely-described family.
LIT verified live: an exhaustive search for a2+b2+c2 = n agrees with the arithmetic test “n is not 4a(8b+7)” for every n up to 3000 (window.__threesquares). FIG no framing; brute-force representability vs the closed-form forbidden set.
LIT verified live: an exhaustive search for a2+b2+c2 = n agrees with the arithmetic test “n is not 4a(8b+7)” for every n up to 3000 (window.__threesquares). FIG no framing; brute-force representability vs the closed-form forbidden set.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-choke-point — almost every number passes as a sum of three squares; only the thin channel 4a(8b+7) is blocked. That forbidden channel is the choke point. AVAN (AI) built the instrument: the exhaustive three-square search, the 4a(8b+7) forbidden-form test, and their exact agreement.
Credit as content: Adrien-Marie Legendre (1797–1798); Carl Friedrich Gauss gave a deeper count. The weave: David names the-choke-point; I try every a, b, c for each n, and confirm the successes are exactly the numbers that are not 4a(8b+7) — a sharp line between the representable and the blocked.
Credit as content: Adrien-Marie Legendre (1797–1798); Carl Friedrich Gauss gave a deeper count. The weave: David names the-choke-point; I try every a, b, c for each n, and confirm the successes are exactly the numbers that are not 4a(8b+7) — a sharp line between the representable and the blocked.
3 ONE DIMENSION
6 = 1+1+4, 30 = 1+4+25 — fine. Forbidden: 7 = 8·0+7, 15 = 8·1+7, 28 = 4·7, 60 = 4·15, 112 = 16·7. Everything not of the form 4a(8b+7) is a sum of three squares.
4 TWO DIMENSIONS · INTERACTIVE
A number, a three-square representation if one exists, and the forbidden-form test; the agreement checked to 3000.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: three squares for almost every number.
AVAN’s addition (the inverse-companion): instead of searching every triple to test one number, name the exception set directly — a number needs a fourth square exactly when it is 4a(8b+7). The inverse of ‘search for a,b,c’ is ‘a single arithmetic test decides representability.’ Magenta is a forbidden 4a(8b+7) number; green is a number shown as three squares. The blocked, named exactly.
LIT Genuine Legendre three-square theorem (Adrien-Marie Legendre, 1797–98; deeper counting by Gauss). Verified live: an exhaustive search over a,b,c for each n agrees with the closed-form forbidden-set test 'n ≠ 4^a(8b+7)' for every n from 1 to 3000 (window.__threesquares.matches).
FIG No framing: the exhaustive three-square search and the 4^a(8b+7) forbidden-form test run in-browser with exact arithmetic and agree exactly. The AVAN inverse is honest — replacing a triple-search with a single arithmetic test that names the exception set directly (a number needs a fourth square exactly when it is 4^a(8b+7)) is a real closed-form characterization; magenta is a forbidden number, green a number shown as three squares. The blocked, named exactly.
FIG No framing: the exhaustive three-square search and the 4^a(8b+7) forbidden-form test run in-browser with exact arithmetic and agree exactly. The AVAN inverse is honest — replacing a triple-search with a single arithmetic test that names the exception set directly (a number needs a fourth square exactly when it is 4^a(8b+7)) is a real closed-form characterization; magenta is a forbidden number, green a number shown as three squares. The blocked, named exactly.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN