Exact arithmetic · three signed volumes · one tiny remainder

Three Mountains Leave Forty-Two

Choose an integer and test the modulo 9 obstruction to writing it as three integer cubes. For 33 and 42, exact BigInt arithmetic expands the published triples and watches three huge signed cubes cancel to the target. Then replay Booker's divisor and perfect-square search certificate, or run the same bounded search idea yourself, with every limit and uncertainty left visible.

The small answer is not estimated from the large terms. Every digit survives. Pick the target first.

Layer one · exact cancellation

One integer enters. Residues and signed cubes come out.

Integer target k

Waiting for live arithmetic.

The three cube terms, width scaled against the largest absolute term

Waiting for exact BigInt terms.

coordinate	integer	signed cube

Source status will appear after computation.

Layer twoThe square hidden inside the search

Verifying a supplied triple is bookkeeping. Finding it is the deeper problem. Booker starts by ordering the coordinates so that |x| ≥ |y| ≥ |z|, then sets d = |x + y|. Since x³ + y³ = (x + y)(x² - xy + y²), the candidate d must divide |k - z³|. One exact square test can then reconstruct the two missing coordinates.

That test is the whole of layer two, so here it is in the open. For the true triple, write d = |x + y| and q = (4|k - z³| - d³) / (3d). Then q = (x - y)². So the row below labelled exact integer sqrt q is not a diagnostic: it is |x - y| itself, and together with d = |x + y| it pins the two missing coordinates exactly. If q is not a perfect square, no triple exists for that (z, d) and the search moves on. Watch the panel: for 42 the exact square root comes out 160974496957893489, which is precisely the difference of the two seventeen-digit coordinates.

Replay a published search certificate

The page derives every row from the selected triple.

Waiting for the live certificate.

Run a bounded search

A Web Worker tests integer pairs (z,d) without freezing the page.

Target k

Absolute z limit

Positive d limit

No bounded search has run yet.

Results will appear here with the exact limits and tested pair count.

A no-hit result is not a nonexistence proof. It means only that this simple browser search found no reconstruction among the displayed (z,d) bounds. The published algorithms add powerful congruence filters and vastly larger distributed computations.

The check · rebuilt in this browser

This panel is filled by the same live arithmetic that drives the instruments. Published values are marked as published. Derived values are recomputed after the page loads.

Current exact triple

coordinate	integer mod 9	cube mod 9

All 27 sums of cube residues

Direct enumeration of base residues 0, 1, and 8 fills this table live. A red result is one of the two unreachable target classes.

Waiting for the finite residue check.

Current search certificate

Machine-checked, over every integer

The obstruction above is enumerated here over all 729 residue triples, which is a statement about residues. The stronger statement, quantified over every integer, is proved by Lean's kernel in research/three-mountains-leave-forty-two/lean/ThreeCubes.lean: cubing commutes with reduction modulo 9, so every integer cube is 0, 1 or 8 mod 9, so no sum of three of them is 4 or 5 mod 9. The file imports nothing, no Mathlib and no Batteries, so the only trusted component is the type checker itself. Its axiom footprint is printed on every typecheck and reads [propext, Quot.sound], with no sorryAx and no Classical.choice. The same file kernel-checks both seventeen-digit identities in exact integer arithmetic, and those two depend on no axioms at all. Reproduce with bash research/three-mountains-leave-forty-two/lean/verify.sh.

Uncertainties, approximations, conventions, and free choices

First known does not mean smallest. The cited papers identify the displayed triples as first known representations. This page does not claim that no smaller representation exists.
The modulo 9 test is necessary, not sufficient. Residues 4 and 5 are impossible for a sum of three integer cubes. Passing the test does not produce a representation.
The seven-case list is a dated literature snapshot. The 2021 peer-reviewed paper lists 114, 390, 627, 633, 732, 921, and 975 as unresolved among eligible positive targets at most 1,000. A web check through 29 July 2026 found no later primary paper or researcher announcement resolving one of them. A preprint revised 22 March 2026 specifically reports an unsuccessful effort on 114. Absence searches can miss sources and this status can become stale.
The general questions remain open. Neither cited paper proves that every eligible integer has one representation, or the stronger expectation that every eligible integer has infinitely many.
Arithmetic convention. All coordinates, cubes, divisibility checks, square roots, and totals use signed JavaScript BigInt integers. The modulo operation is normalized to residues 0 through 8. There are no physical units.
Ordering and duplicates. Certificates order coordinates by descending absolute value. Permuting a triple does not create a new mathematical representation.
Visual scale is a free choice. Bar widths use each cube's exact absolute value divided by the largest absolute cube, rounded only for CSS pixels. The printed arithmetic, not the bar width, is the check.
The browser search is deliberately small. Its user-selected bounds, simple divisor loop, and stop after 20 displayed hits are engineering choices. A no-hit result makes no claim outside that finite rectangle.

Sources and what they support

Andrew R. Booker, “Cracking the problem with 33”, Research in Number Theory 5, article 26, published 19 July 2019. DOI: 10.1007/s40993-019-0162-1. arXiv:1903.04284. peer-reviewed Source for the published 33 triple, the factorization method, and formula (3) used by the certificate instrument.
Andrew R. Booker and Andrew V. Sutherland, “On a question of Mordell”, Proceedings of the National Academy of Sciences 118(11), e2022377118, published online 10 March 2021 and in the 16 March 2021 issue. DOI: 10.1073/pnas.2022377118. arXiv:2007.01209. peer-reviewed Source for the published 42 triple and the seven unresolved eligible targets at most 1,000 reported in section 5.
University of Bristol, “Sum of three cubes for 42 finally solved”, press release issued 6 September 2019. announcement Independent institutional display of the signed 42 coordinates and the announcement date. It is not the peer-reviewed source.
Andrew V. Sutherland, “Sums of cubes”, MIT Mathematics researcher page, checked 29 July 2026. researcher page Independent display of the 42 representation.
Jon Grantham and P. G. Walsh, “Representing integers as a sum of three cubes”, arXiv:2211.12149, version 1, submitted 22 November 2022. preprint Later primary research located during the status check. It says their effort on k = 114 remained unsuccessful. It does not prove nonexistence.

Offline differential verifier: node research/three-mountains-leave-forty-two/verify-three-mountains-leave-forty-two.mjs. It extracts the shipped arithmetic engine from this HTML, sweeps it against an independently written reference implementation, checks the published triples and certificates, and proves that a deliberate mutation is detected.