Six problems a language model answers confidently and wrongly, beside what the engine computes.
An assistant answering from pattern is wrong just often enough to matter, and wrong in a particular way: the answer has the right shape. The series has the right first terms; the integral is a famous value with the wrong limits attached; the factorization stops at the point where inspection runs out. Each row below shows the pattern's answer, the term where it fails, and what the engine returns when it computes instead.
No model is named on this page. The left column is what pattern-matching produces on these problems in general, and it dates the moment any one model improves; the right column is re-verified against the engine on every build, and this page cannot ship if it drifts.
The wrong term: 31x^9/2835
verified against the engine at build
The first four coefficients are common enough to be recalled correctly; the fifth is not. The true coefficient is 62/2835. A series that is right to four terms and wrong at the fifth looks like knowledge and is the shape of a guess.
The wrong term: -1
verified against the engine at build
A 4x4 characteristic polynomial has enough terms for one sign to go wrong on the way to the answer, and a wrong sign produces a wrong eigenvalue that still looks like one. Here the matrix decouples into two 2x2 blocks; the engine finds all four exactly rather than approximating the quartic.
The wrong term: Pi
verified against the engine at build
The Dirichlet integral is famous enough to be recalled, and famous enough to be recalled with the wrong limits: the value over (-inf, inf) is Pi, over (0, inf) it is Pi/2. Both facts are true; attaching the right one to the limits written down is the computation.
The wrong term: 329218123329 declared prime
verified against the engine at build
Dividing out the 3 is easy; deciding whether the twelve-digit cofactor is prime is not something inspection can do, so a plausible answer stops there and calls it prime. The engine factors it, and it is not.
The wrong term: 1
verified against the engine at build
The theorem is real and the shape of the problem invites it, but 10^6 is not 10^9+6. The correct residue is reached only by actually reducing, which is a few hundred squarings - trivial for an engine and impossible by inspection.
The wrong term: Pi^4/96
verified against the engine at build
zeta(2) = Pi^2/6 is remembered by everyone; zeta(4) is remembered as 'Pi^4 over something', and the something is where a recalled answer diverges from a computed one. 90 is right; 96 and 45 both appear in nearby identities and are both wrong here.
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