Qurak

Munasabah tidak sama dengan betul

Enam masalah yang dijawab model bahasa dengan yakin dan salah, di sebelah apa yang dikira enjin.

Pembantu yang menjawab daripada corak tersilap cukup kerap untuk memberi kesan, dan tersilap dengan cara yang tersendiri: jawapannya berbentuk betul. Sirinya mempunyai sebutan awal yang betul; kamirannya ialah nilai terkenal dengan had yang salah dilekatkan; pemfaktorannya berhenti di titik pemerhatian tidak lagi mencukupi. Setiap baris di bawah menunjukkan jawapan corak, sebutan tempat ia gagal, dan apa yang dikembalikan enjin apabila ia mengira.

Tiada model yang dinamakan di halaman ini. Lajur kiri ialah apa yang dihasilkan pemadanan corak pada masalah-masalah ini secara umum, dan ia menjadi lapuk sebaik sahaja mana-mana satu model bertambah baik; lajur kanan disahkan semula terhadap enjin pada setiap binaan, dan halaman ini tidak boleh dikeluarkan jika ia menyimpang.

Series[Tan[x], {x, 0, 9}]

Model bahasa, menjawab daripada corak

x + x^3/3 + 2x^5/15 + 17x^7/315 + 31x^9/2835 + O(x^11)

Sebutan yang salah: 31x^9/2835

Qurak, mengira

SeriesData[x, 0, {1, 0, 1/3, 0, 2/15, 0, 17/315, 0, 62/2835}, 1, 10, 1]

disahkan terhadap enjin semasa binaan

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.

Kira sendiri

Eigenvalues[{{2, 0, 0, 1}, {0, 3, 1, 0}, {0, 1, 3, 0}, {1, 0, 0, 2}}]

Model bahasa, menjawab daripada corak

{4, 2, 3, 1} - or, from a sign slip in the characteristic polynomial, {4, 3, 2, -1}

Sebutan yang salah: -1

Qurak, mengira

{4, 3, 2, 1}

disahkan terhadap enjin semasa binaan

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.

Kira sendiri

Integrate[Sin[x]/x, {x, 0, Infinity}]

Model bahasa, menjawab daripada corak

Pi - the integral over the whole real line, quoted for the half line

Sebutan yang salah: Pi

Qurak, mengira

Pi/2

disahkan terhadap enjin semasa binaan

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.

Kira sendiri

FactorInteger[987654321987]

Model bahasa, menjawab daripada corak

{{3, 1}, {329218123329, 1}} - the cofactor declared prime by inspection

Sebutan yang salah: 329218123329 declared prime

Qurak, mengira

{{3, 1}, {329281, 1}, {999809, 1}}

disahkan terhadap enjin semasa binaan

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.

Kira sendiri

PowerMod[3, 1000000, 1000000007]

Model bahasa, menjawab daripada corak

1 - by Fermat's little theorem, misapplied: 3^(p-1) is 1 mod p, but the exponent here is not p-1

Sebutan yang salah: 1

Qurak, mengira

64935414

disahkan terhadap enjin semasa binaan

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.

Kira sendiri

Sum[1/n^4, {n, 1, Infinity}]

Model bahasa, menjawab daripada corak

Pi^4/96, or Pi^4/45 - a neighbour of the right constant

Sebutan yang salah: Pi^4/96

Qurak, mengira

Pi^4/90

disahkan terhadap enjin semasa binaan

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.

Kira sendiri

Pembantu anda boleh menyerahkan matematik itu kepada enjin

Qurak ialah pelayan MCP. Setelah disambungkan, Claude, Cursor, VS Code atau mana-mana klien MCP mengira dan bukannya meneka - dan ia terbuka pada pelan percuma. Atau cuba satu ungkapan di sini, tanpa akaun.

5 penilaian tanpa nama sejam, bajet 1 saat, matematik sahaja. Akaun menghapuskan ketiga-tiga had itu.