A derivation around a problem by Euler

The problem

Consider the system of equations a + b + c = u², ab + bc + ca = v², abc = w², and derive, step by step, the parametrized solution [a, b, c, u, v, w] = [k²t²(s²−t²)²(s²+t²), k²s²(s²−t²)²(s²+t²), 4k²s⁴t⁴, k(s⁴+t⁴), k²st(s²+t²)(s⁴−t⁴), 2k³s³t³(s²−t²)²(s²+t²)]. I want to understand how one arrives at it.

The underlying problem, first considered by Euler, was about finding “Three integers whose sum, product and the sum of the products of the integers, taken two at a time, are perfect squares”.

What follows is the same problem given to ExaktAI and to the AI assistants built into Mathematica and Maple, the two Computer Algebra Systems (CAS).

ExaktAI: a validated and reproducible derivation

The problem tackled with ExaktAI: Codex as the AI and Mathematica 14.3 as the CAS validating tool. The left panel is the ExaktAI App in Presentation mode, an optional frameless display omitting visual distractions such as the Run ExaktAI button that ran the computation. This mode presents the validated derivation cleanly, and fits all eight steps in a single screenshot. To the right is the Mathematica notebook automatically generated and opened by the ExaktAI App.

ExaktAI on the polynomial system, side by side: on the left, the validated derivation presented step by step in the ExaktAI app; on the right, the executable Mathematica notebook it runs, every step an executed instruction with an output, chained from the equations to the assembled solution — ExaktAI (Codex + Mathematica 14.3): the validated derivation that resulted in the ExaktAI App from running the problem (left) and the executable notebook filled with the validated steps of the solution, where the derivation can be reproduced.

ExaktAI builds the solution from the equations. It writes a + b = x² and c = y², recognizes a Pythagorean structure x² + y² = u² and parametrizes the triple; treats v² as a second Pythagorean relation; forms the quadratic discriminant whose roots are a and b; takes its square root on a stated branch; solves the two linear equations for a and b; recovers c and w; and assembles [a, b, c, u, v, w].

Every line is a reproducible Mathematica input/output, each step references the results of the steps before it, and the final output is the requested derivation, validated.

Wolfram’s 14.3 AI Assistant: not a derivation

The same problem asking for a derivation step-by-step to the Notebook Assistant in Mathematica 14.3, AI model ‘Wolfram’ (the default), persona Code Assistant.

Wolfram Notebook Assistant (Code Assistant persona): a prose narrative that restates the given answer, never computes v, mislabels u as u-squared and w as w-squared, and offers an unexecuted verification script — Notebook Assistant in Mathematica 14.3 (AI model ‘Wolfram’, persona Code Assistant): an AI narrative that verifies rather than derives, the verification the prompt excluded, then a verification script.

In the run shown:

In Step 1, the parameter s renders as “SFailed”, a failed substitution surfaced in the narrative.
In Step 2, a, b, c are taken verbatim from the solution supplied in the prompt (“the expressions provided in the parametrized solution”).
In Step 3, it restates the problem’s defining equations. The answer is assumed, not derived.
In Step 4, it presents u² = k(s⁴+t⁴). Correct is u² = k²(s⁴+t⁴)².
In Step 5, it does not compute v², stating only that the terms “will be polynomials… compute and simplify.”
In Step 6, it presents w² = 2k³s³t³(s²−t²)²(s²+t²). Correct is w² = 4k⁶s⁶t⁶(s²−t²)⁴(s²+t²)².
Asked for the code, it offers a verification script where the problem given and its solution are presented with no code to derive the latter from the former.

None of these are flagged. The AI narrative does not show a derivation, cannot be executed, and is not reproduced by the code provided. The gap is not that Mathematica or the Wolfram AI model are weak; it is that nothing connects “what the AI claims” to what Mathematica could verify.

ExaktAI: an alternative, equivalent derivation

ExaktAI with Codex as the AI and Maple 2026.1 as the CAS validating tool, producing an alternative derivation in 13 steps. For presentation purposes here, the ExaktAI App (left) shows the Problem and steps 1 to 6 out of the 13 it holds, and the Maple document ExaktAI generated (right) shows the remaining steps 7 to 13, to display the 13 steps legibly in one image.

ExaktAI on the polynomial system in Maple: the ExaktAI app in presentation mode on the left showing the problem and the first six validated steps, and the generated Maple document on the right showing steps seven to thirteen through the assembled solution — ExaktAI (Codex + Maple 2026.1): the validated 13-step derivation, the App’s presentation showing steps 1 to 6 and the generated Maple document showing 7 to 13.

Every step is an executed Maple instruction with an output, each referencing the results of the steps before it, the whole derivation marked Validated, and the final output is the requested derivation. As with the case of the Mathematica run, the derivation is reproducible in this case by running the Maple document.

Maple’s 2026.1 AI Assistant: a correct narrative, not a computable derivation

The same problem asking for a derivation step-by-step to Maple’s AI Assistant in Maple 2026.1. The AI presents a genuine derivation: force abc to be a square, parametrize R = (s²+t²)X², apply the Pythagorean identity, then scale, arriving at the exact requested [a, b, c, u, v, w].

Maple AI Assistant on the same problem: the later steps of the inserted derivation in the worksheet, reaching the exact assembled solution at line 24, with several chained equalities (lines 14 and 15) evaluating to false — Maple AI Assistant in Maple 2026.1: the inserted derivation, here its later steps through line (24).

In this case, the narrative of a derivation is correct, but the code supposed to reproduce that narrative in Maple is not, resulting in unexpected mistakes:

several inserted equalities evaluate to false (lines 14 and 15 here, and earlier at 3, 6, 7).
earlier in the worksheet, a square placeholder renders as the literal text “EmptySmallSquare” (line 11).

So the narrative of the derivation is right, but what reaches the worksheet is not a computable derivation but a sequence of statements, most asserting a new expression rather than operating on the previous ones, and several with wrong syntax.

The difference

Mapping all the AI reasoned steps into CAS computations, reproducibility, and validation are the what set ExaktAI apart from CAS with AI Assistants, and are the basis for you to be in command of the computation and to trust AI mathematics.

What validation adds → ExaktAI vs CAS AI assistants →

These are single runs of non-deterministic systems (Mathematica 14.3, Maple 2026.1); another run might decompose or err differently. The two in-CAS assistants also differed from each other here: the Notebook Assistant restated the supplied answer and its computed steps carried errors, whereas Maple’s AI Assistant produced an independent, correct derivation whose worksheet, however, was not computable (a sequence of statements, several with wrong syntax). Both software platforms and their AI assistants are actively evolving; the observations describe what each system produced in these specific runs.