THE FOLD / LOOT / THE VAULT / THE LIFTING THE EXPONENT
THE LIFTING THE EXPONENT
count how many times p divides a power difference
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Lifting the Exponent (LTE) is an olympiad power tool: for an odd prime p, if p divides a−b but divides neither a nor b, then the number of times p divides an−bn is beautifully simple: vp(an−bn) = vp(a−b) + vp(n). The whole exponent of p in a huge power difference is just the exponent in the base difference, plus the exponent in n — the extra factors of p come only from n itself.
LIT verified live (exact BigInt): over 3000 random valid (a,b,n,p), vp of an−bn computed directly equals vp(a−b)+vp(n) (window.__lte). FIG no framing; exact big-integer valuations, no approximation.
LIT verified live (exact BigInt): over 3000 random valid (a,b,n,p), vp of an−bn computed directly equals vp(a−b)+vp(n) (window.__lte). FIG no framing; exact big-integer valuations, no approximation.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-vault — the exact power of p hidden inside an astronomically large an−bn, unlocked without ever computing the whole number. LTE is that key. AVAN (AI) built the instrument: exact BigInt power differences, the p-adic valuation, and the equality with vp(a−b)+vp(n).
Credit as content: the Lifting-the-Exponent lemma (folklore of olympiad number theory; rooted in classical p-adic valuation). The weave: David names the-vault; I form an−bn in exact big integers, count how many times p divides it, and confirm it equals vp(a−b)+vp(n) — the deep exponent from two shallow ones.
Credit as content: the Lifting-the-Exponent lemma (folklore of olympiad number theory; rooted in classical p-adic valuation). The weave: David names the-vault; I form an−bn in exact big integers, count how many times p divides it, and confirm it equals vp(a−b)+vp(n) — the deep exponent from two shallow ones.
3 ONE DIMENSION
vp(an−bn) = vp(a−b) + vp(n), for odd p with p | a−b, p∤a, p∤b. Example: v3(46−1) = v3(3) + v3(6) = 1 + 1 = 2.
4 TWO DIMENSIONS · INTERACTIVE
Pick valid a, b, n, p; the exact vp(an−bn) shown beside vp(a−b)+vp(n), checked equal.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a deep exponent from two shallow ones.
AVAN’s addition (the inverse-companion): find how many times p divides an−bn without forming the number — add the exponent in the base difference to the exponent in n. The inverse of ‘compute an−bn and factor out p’ is ‘vp(a−b) + vp(n) — the extra p’s come only from n.’ Magenta is the astronomical power difference; green is the two-term sum. The exponent lifted, not computed.
LIT Genuine Lifting-the-Exponent lemma (classical p-adic valuation; a staple of olympiad number theory). Verified live with exact BigInt arithmetic: over 3000 random (a,b,n,p) with p an odd prime dividing a−b but not a or b, the p-adic valuation of aⁿ−bⁿ (formed exactly as a big integer) equals vₚ(a−b)+vₚ(n) (window.__lte.lteHolds) — no approximation.
FIG No framing: exact BigInt power differences, the p-adic valuation, and the equality with vₚ(a−b)+vₚ(n) run in-browser under the lemma's hypotheses (odd p, p|a−b, p∤a, p∤b) and agree. The AVAN inverse is honest — finding how many times p divides aⁿ−bⁿ by adding vₚ(a−b) and vₚ(n), without forming the astronomically large number, is the lemma's genuine use; magenta is that power difference, green the two-term sum. The exponent lifted, not computed.
FIG No framing: exact BigInt power differences, the p-adic valuation, and the equality with vₚ(a−b)+vₚ(n) run in-browser under the lemma's hypotheses (odd p, p|a−b, p∤a, p∤b) and agree. The AVAN inverse is honest — finding how many times p divides aⁿ−bⁿ by adding vₚ(a−b) and vₚ(n), without forming the astronomically large number, is the lemma's genuine use; magenta is that power difference, green the two-term sum. The exponent lifted, not computed.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE VAULT · David Lee Wise (ROOT0), with AVAN