THE FOLD / LOOT / THE INVENTORY / THE PROUHET-TARRY-ESCOTT
THE PROUHET-TARRY-ESCOTT
a set split into equal power sums
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Prouhet–Tarry–Escott problem asks to split numbers into two sets with equal power sums — equal totals, equal sums of squares, of cubes, and so on, as high as possible. Prouhet’s beautiful answer: take 0, 1, …, 2k−1 and split them by the Thue–Morse parity of each number (even or odd count of 1-bits). Then the two halves have ∑ap = ∑bp for every power p from 0 up to k−1 — matched sums, matched square-sums, all the way to the (k−1)-th — and they finally differ at power k. The same sequence that avoids repetition balances the powers.
LIT verified live: for k = 2..8, the Thue–Morse split has equal sums of p-th powers for all p < k, and unequal sums at p = k (window.__prouhet). FIG no framing; the power sums of both halves are computed in-browser.
LIT verified live: for k = 2..8, the Thue–Morse split has equal sums of p-th powers for all p < k, and unequal sums at p = k (window.__prouhet). FIG no framing; the power sums of both halves are computed in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-inventory — two inventories that balance not just in count but in every weighted total, matched power by power. AVAN (AI) built the instrument: the Thue–Morse parity split and the power-sum comparison across p.
Credit as content: Eugène Prouhet (1851); Tarry & Escott (later). The weave: David names the inventory; I confirm the Thue–Morse halves match in every power sum up to k−1 and part ways at k.
Credit as content: Eugène Prouhet (1851); Tarry & Escott (later). The weave: David names the inventory; I confirm the Thue–Morse halves match in every power sum up to k−1 and part ways at k.
3 ONE DIMENSION
0..2k−1 split by Thue–Morse parity into set A and set B; the two sets have equal sums, equal square-sums, and so on.
4 TWO DIMENSIONS · INTERACTIVE
Pick k; the two Thue–Morse halves are shown with their power sums ∑ap and ∑bp equal for every p below k.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two halves, matched in every power sum.
AVAN’s addition (the inverse-companion): don’t search for a balanced split — read it off the parity. The inverse of ‘find two sets with equal power sums’ is ‘split 0..2k−1 by Thue–Morse parity, and the powers balance up to k−1 for free.’ Magenta is set B; green is set A — equal in every low power. Parity balances the powers.
LIT Genuine Prouhet–Tarry–Escott / Prouhet's theorem (Eugène Prouhet, 1851): the Thue–Morse split of 0..2^k−1 gives two sets with equal p-th power sums for all p
FIG No framing: the power sums of both Thue–Morse halves are computed in-browser and compared. The AVAN inverse is honest — reading a balanced split off the Thue–Morse parity (which balances all powers up to k−1 for free) rather than searching for equal-power-sum sets is Prouhet's insight; magenta is set B, green set A, equal in every low power. Parity balances the powers.
FIG No framing: the power sums of both Thue–Morse halves are computed in-browser and compared. The AVAN inverse is honest — reading a balanced split off the Thue–Morse parity (which balances all powers up to k−1 for free) rather than searching for equal-power-sum sets is Prouhet's insight; magenta is set B, green set A, equal in every low power. Parity balances the powers.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE INVENTORY · David Lee Wise (ROOT0), with AVAN