16 April 2025

Introducing OpenAI o3 and o4-mini

Lagi dimuat…

Update tanggal 10 Juni 2025: OpenAI o3‑pro saiki kasedhiya kanggo pangguna Pro ing ChatGPT, uga ing API kita. Kaya OpenAI o1‑pro, o3‑pro yaiku versi saka model paling cerdas kita, OpenAI o3, sing dirancang supaya mikir luwih suwe lan menehi tanggapan paling andal. Rincian lengkap bisa ditemokake ing cathetan rilis⁠(mbukak ing jendhela anyar) kita.

Dina iki, kita ngrilis OpenAI o3 lan o4-mini, sing paling anyar ing seri model o-series kita sing dilatih kanggo mikir luwih suwe sadurunge nanggapi. Iki yaiku model paling pinter sing tau kita rilis nganti saiki, makili lompatan gedhe ing kemampuan ChatGPT kanggo kabeh wong saka pangguna penasaran nganti peneliti tingkat lanjut. Kanggo pisanan, model nalar kita bisa nggunakke lan nggabungake saben alat ing ChatGPT kanthi cara ala agen—iki kalebu nelusuri web, nganalisis file sing diunggah lan data liyane nganggo Python, nalar jero babagan input visual, lan malah ngasilake gambar. Sing penting, model-model iki dilatih kanggo nalar kapan lan kepriye nggunakke alat supaya bisa ngasilake jawaban rinci lan dipikir kanthi apik ing format output sing pas, biasane kurang saka semenit, kanggo ngrampungake masalah sing luwih rumit. Iki ngidini model-model iki nangani pitakon multi-aspek kanthi luwih efektif, langkah menuju ChatGPT sing luwih agen lan bisa ngeksekusi tugas kanthi mandiri kanggo sampeyan. Gabungan kekuwatan nalar state-of-the-art karo akses alat lengkap iki ngasilake kinerja sing luwih kuwat kanthi signifikan ing benchmark akademik lan tugas donya nyata, netepake standar anyar kanggo intelijensi lan kegunaan.

Apa sing owah

OpenAI o3 yaiku model nalar paling kuat sing nyurung wates tercanggih ing coding, matematika, sains, persepsi visual, lan liya-liyane. Model iki nyetel SOTA anyar ing benchmark kalebu Codeforces, SWE-bench (tanpa mbangun scaffold khusus model), lan MMMU. Iki cocog kanggo pitakon rumit sing mbutuhake analisis multi-aspek lan jawabané bisa uga ora langsung cetha. Kinerjane kuwat banget utamane ing tugas visual kaya nganalisis gambar, bagan, lan grafis. Ing evaluasi dening ahli eksternal, o3 nggawe 20 persen luwih sithik kesalahan gedhe tinimbang OpenAI o1 ing tugas angel ing donya nyata—utamane unggul ing area kaya pemrograman, bisnis/konsultasi, lan ideasi kreatif. Penguji awal nyorot ketegasan analitisé minangka mitra mikir lan nekanake kemampuané kanggo ngasilake lan ngevaluasi kanthi kritis hipotesis anyar—utamane ing konteks biologi, matematika, lan teknik.

OpenAI o4-mini yaiku model sing luwih cilik sing dioptimalake kanggo nalar cepet lan efisien biaya—model iki nggayuh kinerja luar biasa kanggo ukuran lan biayane, utamane ing matematika, coding, lan tugas visual. Iki model benchmark kanthi kinerja paling apik ing AIME 2024 lan 2025. Sanajan akses menyang komputer kanthi nyata nyuda tingkat angelé ujian AIME, kita uga nemokake yen o4-mini nggayuh 99.5% pass@1 (100% consensus@8) ing AIME 2025 nalika diwenehi akses menyang interpreter Python. Sanajan asil iki ora kudune dibandhingake karo kinerja model tanpa akses alat, iki salah siji conto carane efektif o4-mini nggunakke alat sing kasedhiya; o3 nuduhake peningkatan sing padha ing AIME 2025 saka panggunaan alat (98.4% pass@1, 100% consensus@8).

Ing evaluasi ahli, o4-mini uga ngluwihi pendahulune, o3‑mini, ing tugas non-STEM uga domain kaya data science. Amarga efisiensine, o4-mini ndhukung wates panggunaan sing luwih dhuwur tinimbang o3, dadi pilihan kuwat kanggo volume gedhe lan kapasitas pangolahan dhuwur kanggo pitakon sing entuk manfaat saka nalar. Evaluator ahli eksternal menehi rating yen kaloro model nuduhake paningkatan ing ngetutake instruksi lan tanggapan sing luwih migunani lan bisa diverifikasi tinimbang pendahulune, amarga intelijensi sing luwih apik lan kalebu sumber web. Dibandhingake karo iterasi sadurunge saka model nalar kita, loro model iki uga bakal krasa luwih alami lan obrolan, utamane nalika ngrujuk memori lan obrolan kepungkur supaya tanggapan luwih personal lan relevan.

Multimodal

Coding

Kabeh run evaluasi SWE-bench nggunakke subset tetep n=477 tugas terverifikasi sing wis divalidasi ing infrastruktur internal kita.

Ngetutake instruksi lan panggunaan alat ala agen

Kabeh model dievaluasi ing setelan ‘upaya nalar’ dhuwur—padha karo varian kaya ‘o4-mini-high’ ing ChatGPT.

Terus ngembangake sinau penguatan

Sajrone pangembangan OpenAI o3, kita ngamatake yen sinau penguatan skala gedhe nuduhake tren sing padha karo “komputasi luwih akeh = kinerja luwih apik” sing diamati ing pretraining seri GPT. Kanthi ngulangi jalur penskalaan—saiki ing RL—kita wis nyurung tambahan siji orde magnitudo ing komputasi latihan lan nalar wektu inferensi, nanging isih ndeleng paningkatan kinerja sing cetha, mbuktekake yen kinerja model terus mundhak nalika diwenehi wektu luwih akeh kanggo mikir. Ing latensi lan biaya sing padha karo OpenAI o1, o3 menehi kinerja luwih dhuwur ing ChatGPT—lan kita wis verifikasi yen yen diwenehi mikir luwih suwe, kinerjane terus munggah.

Kita uga nglatih loro model iki supaya nggunakke alat liwat sinau penguatan—mulang model ora mung carane nggunakke alat, nanging uga nalar kapan kudu nggunakke. Kemampuané kanggo masang alat adhedhasar asil sing dikarepake ndadekake luwih mumpuni ing situasi sing mbukak—utamane sing melu nalar visual lan alur kerja multi-langkah. Paningkatan iki katon ing benchmark akademik lan tugas donya nyata, kaya sing dilapurake para penguji awal.

Mikir nganggo gambar

Kanggo pisanan, model-model iki bisa nggabungake gambar langsung menyang rantai pamikirané. Ora mung ndeleng gambar—nanging mikir bareng gambar kuwi. Iki mbukak kelas pemecahan masalah anyar sing nyawijikake nalar visual lan tekstual, sing katon saka kinerja state-of-the-art ing benchmark multimodal.

Wong bisa ngunggah foto papan tulis, diagram buku teks, utawa sketsa tangan, lan model bisa napsirake—sanajan gambare burem, kebalik, utawa kualitase kurang apik. Kanthi panggunaan alat, model bisa ngolah gambar kanthi langsung—muter, ngzoom, utawa ngowahi minangka bagean saka proses nalaré.

Model-model iki menehi akurasi paling apik ing kelasé kanggo tugas persepsi visual, supaya bisa ngrampungake pitakon sing sadurunge ora bisa digayuh. Delengen blog riset nalar visual⁠ kanggo sinau luwih akeh.

Menyang panggunaan alat ala agen

OpenAI o3 lan o4-mini nduweni akses lengkap menyang alat ing ChatGPT, uga alat kustom sampeyan dhewe liwat nelpon fungsi ing API. Model-model iki dilatih kanggo nalar babagan cara ngrampungake masalah, milih kapan lan kepriye nggunakke alat kanggo ngasilake jawaban rinci lan dipikir kanthi apik ing format output sing pas kanthi cepet—biasane kurang saka semenit.

Contone, pangguna bisa takon: “Kepiye panggunaan energi musim panas ing California dibandhingake taun kepungkur?” Model bisa nelusuri web kanggo data utilitas publik, nulis kode Python kanggo mbangun prakiraan, ngasilake grafik utawa gambar, lan nerangake faktor utama ing balik prediksi kasebut, kanthi ngrangkai pirang-pirang panggilan alat. Nalar ngidini model nanggapi lan ngowahi arah yen perlu adhedhasar informasi sing ditemoni. Contone, model bisa nelusuri web kaping pirang-pirang kanthi pitulungan panyedhiya telusur, ndeleng asil, lan nyoba telusur anyar yen butuh info luwih akeh.

Pendekatan sing luwes lan strategis iki ngidini model nangani tugas sing mbutuhake akses menyang informasi paling anyar ngluwihi kawruh bawaan model, nalar sing luwih dawa, sintesis, lan generasi output lintas modalitas.

Kabeh conto dirampungake nganggo OpenAI o3.

OpenAI o3

Solve the following very hard math problem using your in-built knowledge of the literature:

Construct a degree 19 polynomial \ (p(x)) \ in \ mathbb{C}[x] such that \ (X := \ {\, p(x) = p(y) \, \} \ subset \ mathbb{P}^{1} \ times \ mathbb{P}^{1}\ ) has at least 3 (but not all linear) irreducible components over \ mathbb{C}. Choose \ (p(x)\ ) to be odd, monic, have real coefficients and linear coefficient \ (-19\ ) and calculate \ (p(19)\ ).

Nalar babagan pambentukan polinomial sajrone 55 detik

The curve defined by p(x) = p(y) factors if p(x) is decomposable or has repeated factors. Actually, p(x)-p(y) always factors, and it’s divisible by (x-y). This gives one factor, and the rest is a degree d-1 polynomial in x and y. For the curve to have at least three irreducible components, p(x)-p(y) must factor into at least three nontrivial factors. Each factor corresponds to a component in projective space, and at least one must be nonlinear.
I’m checking the Dickson polynomials D_n(x,a) and focusing on their coefficients. With a=1, it seems the leading coefficient is always 1, confirming that Dickson polynomials are monic. The linear coefficient for odd n is following a predictable sign pattern: for n=19, it gives a coefficient of -19, which is exactly what I need. So the Dickson polynomial D_n(x,1) with odd n fits nicely with the linear coefficient I was looking for.
I've confirmed that the linear coefficient for p(x)=D19(x)p(x) = D_{19}(x)p(x)=D19(x) is correct, with the necessary scaling factor to match. Now for the next step, I need to compute D19(19)D_{19}(19)D19(19). Since Tn(z)T_n(z)Tn(z) grows quickly for real z>1z > 1z>1, I plan to compute this value using Python, leveraging the recurrence to avoid mistakes—it's going to be a big number. I'll make sure to present the numeric result neatly so it's understandable since the coefficients are integers.
The Dickson polynomial D_{19}(x,1) is confirmed to have the required properties: monic, odd, real coefficients, and a linear coefficient of -19. The factorization of D_{19}(x) into irreducible components is also verified and satisfies the condition of having at least 10 components (1 linear + 9 quadratics). To compute p(19), we'll use the recurrence relation for Dickson polynomials, but there's also a closed-form expression involving cosine that could be handy here. I'll proceed accordingly!
I confirmed the linear coefficient is working correctly, which means now I need to calculate p(19)=D19(19)p(19) = D_{19}(19)p(19)=D19(19). This involves evaluating T19(9.5)T_{19}(9.5)T19(9.5), which grows rapidly. To ensure I get an accurate result, I'll compute this using recurrence in Python. The result is a large number: 1,876,572,071,974,094,803,391,179. I'll use the commas to make it easier to read: p(19)=1,876,572,071,974,094,803,391,179p(19) = 1,876,572,071,974,094,803,391,179p(19)=1,876,572,071,974,094,803,391,179.
For a = 1, there’s a relationship between Dickson polynomials and Chebyshev polynomials: D_n(x, 1) = 2^(n-1) T_n(x/2). Checking this, we see that the definition of D_n(u + 1/u) aligns well with Chebyshev’s formula. By setting x = 2 cos(θ), we find that D_n(x) equals 2 T_n(x/2). Importantly, the leading coefficient works out to 1, confirming the identity holds. This is nice!
Now, I need to verify that p(x)−p(y)p(x) - p(y)p(x)−p(y) factors as desired, relying on the known Dickson polynomial factorization. According to the literature (like Fried and Jarden), the difference Dn(x)−Dn(y)D_n(x) - D_n(y)Dn(x)−Dn(y) factors into a product involving (x−y)(x - y)(x−y) and other expressions. I'll provide the explicit polynomial D19(x,1)D_{19}(x, 1)D19(x,1), list the coefficients, and explain how it satisfies the desired properties: monic, odd, etc. Lastly, I'll compute p(19)p(19)p(19).

Analyzed

Python

1from math import comb, cos, acos
2# Let's compute Dickson D_n(x,a=1) recursively
3def dickson(n,x,a=1):
4    if n==0:
5        return 2
6    if n==1:
7        return x
8    Dm2=2
9    Dm1=x
10    for k in range(2,n+1):
11        D= x*Dm1 - a*Dm2
12        Dm2, Dm1 = Dm1, D
13    return Dm1
14
15d19 = dickson(19,19,1)
16d19

1876572071974094803391179

Analyzed

Python

1d19

1876572071974094803391179

Analyzed

Python

1import locale, sys, math, json, textwrap, re, decimal

Analyzed

Python

1"{:,}".format(d19)

'1,876,572,071,974,094,803,391,179'

Analyzed

Python

1# compute coefficients of Dickson polynomial D_n(x,1) via recurrence polynomial representation
2from sympy import symbols, Poly
3
4x = symbols('x')
5def dickson_poly(n):

Poly(x**19 - 19*x**17 + 152*x**15 - 665*x**13 + 1729*x**11 - 2717*x**9 + 2508*x**7 - 1254*x**5 + 285*x**3 - 19*x, x,

Analyzed

Python

1poly19.as_expr()

x**19 - 19*x**17 + 152*x**15 - 665*x**13 + 1729*x**11 - 2717*x**9 + 2508*x**7 - 1254*x**5 + 285*x**3 - 19*x

Done

A convenient way to force many factors in
$p(x) \;-\; p(y)$ is to use a Dickson polynomial of the first kind.
For a parameter $a \in \mathbb{C}$ in math these are defined by

$D_0\!\left(x,\, a\right) = 2,\; D_1\!\left(x,\, a\right) = x,\; D_n\!\left(x,\, a\right) = x\,D_{n-1}\!\left(x,\, a\right) - a\,D_{n-2}\!\left(x,\, a\right)\quad (n \ge 2),$

and satisfy the functional identity

$D_n\!\left(u + \frac{a}{u},\, a\right) = u^{n} + \frac{a^{n}}{u^{n}}.$

1. Choosing the polynomial

Take $a = 1$ and $n=19$ .
Because $n$ is odd, $D_{19}\!\left(x,\, 1\right)$ is an odd polynomial; the recurrence immediately shows it is monic and has integer (hence real) coefficients.
The coefficient of $x$ is $x = (-1)^{\frac{n-1}{2}}\,n = -19,$ so all requested normalisations are satisfied.

Writing it out gives

$\boxed{% p(x) = D_{19}\!\left(x,1\right) = x^{19} - 19x^{17} + 152x^{15} - 665x^{13} + 1729x^{11} - 2717x^{9} + 2508x^{7} - 1254x^{5} + 285x^{3} - 19x% }\,.$

2. Irreducible components of $fX \;=\; \bigl\{\, p(x) = p(y) \,\bigr\}$

For Dickson polynomials one has the classical factorisation

$D_n\!\left(x,1\right) \;-\; D_n\!\left(y,1\right) \;=\; (x - y)\, \prod_{j=1}^{\frac{n-1}{2}} \Bigl( x^{2} + y^{2} - 2\cos\!\bigl(\tfrac{2\pi j}{n}\bigr)\,xy - 4\sin^{2}\!\bigl(\tfrac{\pi j}{n}\bigr) \Bigr),$

where each quadratic factor is irreducible over $\mathbb{C}$ (see, e.g., Lidl–Niederreiter, Finite Fields, §3.6).
For $n=19$ this gives

one linear component $x=y;$
nine quadratic components (corresponding to $\bigl(j = 1,\,\ldots,\,9\bigr).$

Hence $X \subset \mathbb{P}^{1} \times \mathbb{P}^{1}$ has $10$ irreducible components, at least three of them and not all linear, as required.

3. Evaluating $p(19)$

Using the recurrence (or the identity $D_n\!\left(x,\,1\right) \;=\; 2\,T_n\!\bigl(\tfrac{x}{2}\bigr)$ with Chebyshev $T_n$ one obtains

$p(19) \;=\; D_{19}\!\left(19,\,1\right) \;=\; 1\,876\,572\,071\,974\,094\,803\,391\,179.$

Thus the polynomial above fulfils every condition, and

$\boxed{% p(19) \;=\; 1{,}876{,}572{,}071{,}974{,}094{,}803{,}391{,}179 }.$

OpenAI o1

Solve the following very hard math problem using your in-built knowledge of the literature:

Mikir sajrone 1m 19d

I’m piecing together a degree 19 polynomial with only odd powers and a linear coefficient of -19, like: p(x)=x19+a17x17+…+a1xp(x) = x^{19} + a_{17}x^{17} + \ldots + a_1xp(x)=x19+a17x17+…+a1x.

I'm examining if the polynomial p(x)=x19+a17x17+…+a3x3−19xp(x) = x^{19} + a_{17}x^{17} + \ldots + a_3x^3 - 19xp(x)=x19+a17x17+…+a3x3−19x has a notable classical form, considering symmetry or factorization for multiple irreducible components.

To ensure p(x)p(x)p(x) has three or more irreducible components, I’m considering the factorization of p(x)−p(y)p(x) - p(y)p(x)−p(y) into at least three irreducible factors in C[x,y]\mathbb{C}[x,y]C[x,y].

I'm working through x19−y19x^{19} - y^{19}x19−y19 to ensure it balances linear and nonlinear factors, aiming for a polynomial with at least three irreducible components that aren't all linear.

Taking a closer look at how p(x)−p(y)p(x) - p(y)p(x)−p(y) can be factored into three distinct polynomials in xxx, aiming to achieve the desired irreducible components.

I'm thinking about how composing ppp with simpler polynomials, like xm−cx^m - cxm−c, could result in multiple irreducible components, achieving at least three distinct factors.

I’m exploring the factorization of p(x)−p(y)p(x) - p(y)p(x)−p(y) with cyclotomic polynomials. This method generates nnn linear irreducible factors, which might not achieve the required three distinct components.

I'm focusing on constructing an "odd" and monic polynomial, ensuring all exponents are odd and the leading term is x19x^{19}x19. This approach aims to achieve the desired factorization.

I'm working with the Chebyshev polynomials to ensure p(x)−p(y)p(x) - p(y)p(x)−p(y) has at least three irreducible components, leveraging their even or odd properties to manage coefficients.

Considering the odd Chebyshev polynomial T19(x)T_{19}(x)T19(x) for potential factorization, while also examining composite polynomials of lower degrees to enhance factorization properties.

I'm analyzing how a polynomial composed of lower-degree polynomials, like (x3−3x)k(x^3 - 3x)^k(x3−3x)k, could yield p(x)−p(y)p(x) - p(y)p(x)−p(y) with at least three irreducible components.

Piecing together polynomials with integer coefficients, like (x3−3x)k(x^3 - 3x)^k(x3−3x)k, to achieve factorization in p(x)−p(y)p(x) - p(y)p(x)−p(y) with at least three irreducible components, aiming for degree 19.

I’m examining how p(x)=(x3−3x)kp(x) = (x^3 - 3x)^kp(x)=(x3−3x)k or p(x)=x19−19xp(x) = x^19 - 19xp(x)=x19−19x might yield p(x)−p(y)p(x) - p(y)p(x)−p(y) with three irreducible components, focusing on ensuring the polynomial is of degree 19.

I’m analyzing p(x)−p(y)p(x) - p(y)p(x)−p(y) into (x−y)(x18+x17y+…+y18−19)(x-y)(x^{18} + x^{17}y + \ldots + y^{18} - 19)(x−y)(x18+x17y+…+y18−19) to identify irreducible components. The polynomial factor x18+x17y+…+y18−19x^{18} + x^{17}y + \ldots + y^{18} - 19x18+x17y+…+y18−19 remains challenging to factor further.

I’m thinking through a new approach: adjusting the Chebyshev polynomial T19(x)T_{19}(x)T19(x) to make it monic, aiming to achieve at least three irreducible components in p(x)−p(y)p(x) - p(y)p(x)−p(y).

OK, let's think through. I’m tweaking T19(x)T_{19}(x)T19(x)'s coefficients to keep the polynomial monic and check if it simplifies into multiple irreducible components.

I'm thinking over various factorization strategies for a monic polynomial of degree 19. The goal is to ensure at least one non-linear factor and achieve at least three irreducible components in p(x)−p(y)p(x) - p(y)p(x)−p(y).

I’m updating my approach to factor x2+xy+y2−1x^2 + xy + y^2 - 1x2+xy+y2−1 over complex numbers by exploring change of variables and third roots of unity. This helps in achieving the desired polynomial structure.

I'm combining x3−xx^3 - xx3−x with x19−19xx^{19} - 19xx19−19x to tweak the polynomial's coefficients and complexity, aiming to include more non-linear factors for the desired irreducibility.

Piecing together p(x)=x19−x3−19xp(x) = x^{19} - x^3 - 19xp(x)=x19−x3−19x to ensure it’s odd, monic, and has the required linear coefficient. This targets the desired irreducible factorizations.

I’m mapping out the second factor x18+…+y18−x2−xy−y2−19x^{18} + \ldots + y^{18} - x^2 - xy - y^2 - 19x18+…+y18−x2−xy−y2−19 to see if it can split into three irreducible components.

I’m considering methods to factorize the polynomial by making it divisible by x−yx - yx−y and ensuring the remaining factor splits into exactly two irreducible components, aiming for a less complex factorization.

I’m examining the polynomial x18+x17y+…+y18x^{18} + x^{17}y + \ldots + y^{18}x18+x17y+…+y18 and noting it simplifies to x19−y19x−y\frac{x^{19} - y^{19}}{x - y}x−yx19−y19, which might reveal further irreducible factors.

I’m mapping out the polynomial x(x18−1)−y(y18−1)x−y\frac{x(x^{18}-1) - y(y^{18}-1)}{x-y}x−yx(x18−1)−y(y18−1) for further factorization. This might yield just two irreducible components, which falls short of three.

I’m exploring whether the polynomial p(x)=x19−(x3)p(x) = x^19 - (x^3)p(x)=x19−(x3) can be expressed as the composition of a smaller polynomial with xkx^kxk, potentially revealing multiple irreducible components.

I'm examining if the polynomial T19(x)T_{19}(x)T19(x) can be factored using trigonometric identities, seeking partial factorization without fully linearizing, to gain multiple irreducible components.

OK, let me see. Trigonometric identities show multiple solutions for θ\thetaθ and ϕ\phiϕ, likely leading to 19 or 38 linear factors for T19(x)T_{19}(x)T19(x).

I’m examining how the equation arccos⁡x=±arccos⁡y+2kπ19\arccos x = \pm \arccos y + \frac{2k\pi}{19}arccosx=±arccosy+192kπ leads to possible multiple branches, potentially irreducible in the algebraic sense, and needing an example with at least three non-linear irreducible components.

OK, let me see. I’m pulling together details on the polynomial x3(x2−1)8+xx^3 (x^2 - 1)^8 + xx3(x2−1)8+x. It’s monic, with x19x^{19}x19 as the leading term, ensuring it meets key criteria.

I’m noting that the polynomial p(x)=x3(x2−1)8+xp(x) = x^3 (x^2 - 1)^8 + xp(x)=x3(x2−1)8+x is odd, with its linear coefficient patterning to -19.

I’m figuring out how to adjust the linear coefficient of polynomial p(x)p(x)p(x) to -19 by adding a constant factor ccc in front of xxx to ensure it’s adequately modified.

By tweaking the linear coefficient of p(x)=x3(x2−1)8+cxp(x) = x^3 (x^2 - 1)^8 + c xp(x)=x3(x2−1)8+cx to -19, it ensures the polynomial remains monic, odd, and real.

I’m working on adding a constant factor ccc to xxx in p(x)=x3(x2−1)8+cxp(x) = x^3 (x^2 - 1)^8 + c xp(x)=x3(x2−1)8+cx to ensure the linear coefficient is -19, making the polynomial monic, odd, and real.

I'm piecing together the polynomial p(x)=x3(x−1)8(x+1)8p(x) = x^3 (x-1)^8 (x+1)^8p(x)=x3(x−1)8(x+1)8 and considering how p(x)−p(y)p(x) - p(y)p(x)−p(y) might factor with factors like x−yx-yx−y.

I’m confirming p(x)p(x)p(x) is monic, odd, and real, with a linear coefficient of -19, by expanding x3(x2−1)8x^3 (x^2 -1)^8x3(x2−1)8 to set the constant ccc.

I found that for p(x) to have a coefficient of -19 for the x term, c must be -19. The polynomial, p(x), incorporates this constant and preserves the original polynomial's structure.

Below is one concrete example (over the reals) of an odd, monic, degree 19 polynomial with linear coefficient $-19$ whose zero-locus relation $\{\,p(x)=p(y)\,\}\ \text{in }\mathbb{P}^{1}\times\mathbb{P}^{1}$ splits into at least three irreducible components (and not all components are linear). We then compute $p(19)$ .

1. The polynomial

Consider

$p(x) \;=\; x^{3}\,\bigl(x^{2}-1\bigr)^{8} \;-\; 19x.$

We can expand

$x^{3}\,\bigl(x^{2}-1\bigr)^{8} \;=\; x^{19}\;-\;8x^{17}\;+\;28x^{15}\;-\;56x^{13}\;+\;70x^{11} \;-\;56x^{9}\;+\;28x^{7}\;-\;8x^{5}\;+\;x^{3}.$

Hence

$p(x) \;=\; x^{19} \;-\; 8x^{17} \;+\; 28x^{15} \;-\; 56x^{13} \;+\; 70x^{11} \;-\; 56x^{9} \;+\; 28x^{7} \;-\; 8x^{5} \;+\; x^{3} \;-\; 19x.$

Checking the required properties

Degree 19, monic:
The leading term is $x^{19}$ with coefficient $1$ .

Odd polynomial:
All exponents are odd $\bigl(19,\,17,\,\ldots,\,3,\,1\bigr),$ and the constant term is $0$ . Indeed,

$p(-x) \;=\; -\,p(x),$

which makes $p$ an odd polynomial.

Real coefficients:
All coefficients are real integers.

Linear coefficient $-19$ :
The coefficient of $x$ is clearly $-19$ .

At least three irreducible components, not all linear:
Over $\mathbb{C}$ , the locus

$X \;=\; \bigl\{\, (x,y) : p(x)=p(y) \,\bigr\} \;\subset\; \mathbb{P}^{1}\times\mathbb{P}^{1}$

factors in part because $p(x)$ has (for instance) the repeated factors $\bigl(x^{2}-1\bigr)^{8}$ in the big piece $x^{3}\,\bigl(x^{2}-1\bigr)^{8}$ . One sees that $\{\,p(x)=p(y)\,\}$ contains (at least) the divisor $${x=y\}$$ plus further factors coming from the roots $x = 0 \;\text{or}\; x = \pm 1,$ ensuring multiple irreducible components (and not all of them are hyperplanes).

2. Computing $p(19)$

A direct substitution $x=19$ gives a large integer. Symbolic expansion or a computer algebra system yields

$p(19) \;=\; 1934999285524070399999639.$

Hence the required value is

\boxed{\,p(19) \;=\; 1934999285524070399999639\,}. \] :contentReference[oaicite:0]{index=0}

OpenAI o3 menehi tanggapan sing bener tanpa nggunakke telusur, dene o1 gagal menehi tanggapan sing bener.

Majokake nalar efisien biaya

Biaya vs kinerja: o3‑mini lan o4-mini

Evaluasi AIME OpenAI o3-mini lan o4-mini

Evaluasi GPQA OpenAI o3-mini lan o4-mini

Biaya vs kinerja: o1 lan o3

OpenAI o3 lan o4-mini yaiku model paling cerdas sing tau kita rilis, lan uga asring luwih efisien tinimbang pendahulune, OpenAI o1 lan o3‑mini. Contone, ing kompetisi matematika AIME 2025, frontier biaya-kinerja kanggo o3 kanthi ketat luwih apik tinimbang o1, lan kanthi cara sing padha, frontier o4-mini kanthi ketat luwih apik tinimbang o3‑mini. Luwih umum maneh, kita ngarepake yen kanggo sebagian besar panggunaan ing donya nyata, o3 lan o4-mini uga bakal luwih pinter lan luwih murah tinimbang o1 lan o3‑mini.

Keamanan

Saben paningkatan kemampuan model mbutuhake paningkatan keamanan sing sepadan. Kanggo OpenAI o3 lan o4-mini, kita mbangun ulang data latihan keamanan kanthi lengkap, nambah prompt penolakan anyar ing area kaya ancaman biologis (biorisk), generasi malware, lan jailbreak. Data anyar iki ndadekake o3 lan o4-mini nggayuh kinerja kuwat ing benchmark penolakan internal kita (umpamane, hirarki instruksi⁠, jailbreak). Saliyane kinerja kuwat kanggo penolakan model, kita uga ngembangake mitigasi tingkat sistem kanggo nandhai prompt mbebayani ing area risiko wates. Kaya karya sadurunge kita ing generasi gambar⁠, kita nglatih monitor LLM nalar sing makarya saka spesifikasi keamanan sing ditulis manungsa lan bisa diinterpretasi. Nalika ditrapake kanggo biorisk, monitor iki kasil nandhai ~99% obrolan ing kampanye human red‑teaming kita.

Kita nguji ketahanan loro model kanthi program keamanan paling ketat nganti saiki. Selaras karo Preparedness Framework⁠ sing dianyari, kita ngevaluasi o3 lan o4-mini ing telung area kemampuan sing dilacak lan dicakup dening Kerangka Kesiapan: biologis lan kimia, keamanan siber, lan perbaikan diri AI. Adhedhasar asil evaluasi iki, kita nemtokake yen o3 lan o4‑mini tetep ana ing ngisor ambang "High" saka Kerangka kasebut ing kabeh telung kategori. Kita wis nerbitake asil rinci saka evaluasi iki ing kertu sistem⁠.⁠

Codex CLI: nalar tercanggih ing terminal

Kita uga nuduhake eksperimen anyar: Codex CLI, agen coding entheng sing bisa sampeyan jalanake saka terminal. Iki bisa langsung makarya ing komputer sampeyan lan dirancang kanggo ngoptimalake kemampuan nalar model kaya o3 lan o4-mini, kanthi dhukungan tambahan sing bakal teka kanggo model API liyane kaya GPT‑4.1⁠.

Sampeyan bisa entuk manfaat saka nalar multimodal saka command line kanthi ngirim screenshot utawa sketsa fidelitas rendah menyang model, digabung karo akses menyang kode sampeyan sacara lokal. Kita nganggep iki minangka antarmuka minimal kanggo nyambungake model-model kita karo pangguna lan komputeré. Codex CLI saiki open-source kanthi lengkap ing github.com/openai/codex⁠(mbukak ing jendhela anyar).

Bebarengan karo iku, kita ngluncurake inisiatif $1 yuta kanggo ndhukung proyek sing nggunakke Codex CLI lan model OpenAI. Kita bakal ngevaluasi lan nampa aplikasi hibah kanthi kenaikan $25.000 USD ing wangun kredit API. Proposal bisa diajukake ing kene.

Akses

Pangguna ChatGPT Plus, Pro, lan Team bakal ndeleng o3, o4-mini, lan o4-mini-high ing pamilih model wiwit dina iki, ngganti o1, o3‑mini, lan o3‑mini‑high. Pangguna ChatGPT Enterprise lan Edu bakal entuk akses sajrone seminggu. Pangguna gratis bisa nyoba o4-mini kanthi milih 'Think' ing composer sadurunge ngirim pitakone. Batas rate ing kabeh paket tetep ora owah saka set model sadurunge.

Kita ngarep-arep bakal ngrilis OpenAI o3‑pro sajrone sawetara minggu kanthi dhukungan alat lengkap. Saiki, pangguna Pro isih bisa ngakses o1‑pro.

o3 lan o4-mini uga kasedhiya kanggo developer wiwit dina iki liwat Chat Completions API lan Responses API (sawetara developer perlu verifikasi organisasi⁠(mbukak ing jendhela anyar) supaya bisa ngakses model-model iki). Responses API ndhukung ringkesan nalar, kemampuan kanggo njaga token nalar ing sekitar nelpon fungsi supaya kinerja luwih apik, lan enggal bakal ndhukung alat bawaan kaya telusur web, telusur file, lan code interpreter ing nalar model. Kanggo miwiti, jelajahi dokumentasi kita⁠⁠(mbukak ing jendhela anyar) lan tunggu kabar anyaran liyane.

Apa sabanjure

Anyaran dina iki nggambarake arah sing dituju model-model kita: kita lagi nyawijikake kemampuan nalar khusus saka o-series karo luwih akeh kemampuan obrolan alami lan panggunaan alat saka GPT‑series. Kanthi nyawijikake kekuwatan iki, model kita ing mangsa ngarep bakal ndhukung obrolan sing lancar lan alami bebarengan karo panggunaan alat proaktif lan pemecahan masalah tingkat lanjut.

Update tanggal 28 Juli 2025: Dataset lan asil SWE-Lancer dianyari per 17 Juli 2025, kasedhiya ing: https://github.com/openai/preparedness⁠(mbukak ing jendhela anyar) lan ing kertu sistem kita. Update iki ngrampungake sawetara masalah sing mengaruhi asil dolar sing dipikolehi lan mbusak syarat konektivitas internet nalika eksekusi, saengga ngilangi sumber variasi utama ing kinerja model.

Update tanggal 16 April 2025: asil kanggo o3 ing Charxiv-r lan Mathvista dianyari kanggo nggambarake owah-owahan system prompt sing ora ana ing evaluasi asli.

Puter ulang livestream

Penulis

OpenAI

Cathetan sikil

* Angka evaluasi tau-bench dirata-rata ing 5 run kanggo nyuda varians, lan dijalanke tanpa alat kustom utawa prompting apa wae. Kita nemokake rollout retail tau-bench luwih rentan marang kesalahan model pangguna. Angka kanggo batang sing diarsir dijalanke nganggo GPT-4.1 minangka model pangguna amarga model iki luwih apik tinimbang GPT-4o ing ngetutake instruksi.

* SWE-bench nggunakke panjang konteks maksimal 256k, sing ningkatake tingkat solve kanggo o4-mini kira-kira 3% lan mengaruhi tingkat solve o3 kurang saka 1%. Kita uga ora kalebu 23 sampel sing ora bisa dijalanke ing infrastruktur internal kita.

** Nalika browsing diaktifake, model kadang bisa nemokake jawaban pas online, contone kanthi maca kiriman blog sing ngemot masalah conto saka dataset. Kita nyuda kekhawatiran model curang nalika browsing nganggo rong strategi:

Domain sing wis kita amati tau dadi sumber model curang ing jaman kepungkur diblokir.
Nggunakake model nalar minangka monitor kanggo mriksa kabeh token ing saben upaya, supaya bisa ngenali prilaku sing curiga. Prilaku curiga ditegesi minangka "kaca, file, utawa cuplikan sing tujuan utamane menehi jawaban pas kanggo pitakon spesifik iki—umpamane, kunci penilaian resmi, gist “solusi” bocor, utawa diskusi sing ngutip jawaban rampung saktemene." Prilaku jinak ditegesi minangka "Sembarang sumber otoritatif sing bisa dirujuk manungsa sing telaten (dokumentasi, manual, makalah ilmiah, artikel bereputasi) sanajan tanpa sengaja ngemot jawaban sing bener." Saben upaya sing dinilai monitor curiga diitung salah. Akèh sampel sing gagal amarga cek iki yaiku masalah sing solusi pasé kasedhiya ing pirang-pirang sumber internet ora ana gandhengane karo HLE.

Pungkasan, evaluasi kita kanthi browsing bisa uga ora sampurna bisa direproduksi ing OpenAI API amarga ana bedane backend search engine antarane ChatGPT lan OpenAI API. Asil iki dimaksudake supaya makili pengalaman pangguna ChatGPT, nanging gumantung panjaluk, konfigurasi telusur bisa owah saka wektu ke wektu.

Kontributor

Aaditya Singh, Aaron Schlesinger, Adam Fry, Adam Lerer, Adam Perelman, Adam Walker, Ahmed El-Kishky, Aidan Clark, Aidan McLaughlin, Aiden Low, Akila Welihinda, Akshay Nathan, Aleksander Madry, Aleksandra Spyra, Alex Karpenko, Alex Neitz, Alex Tachard Passos, Alex Wei, Alexander Prokofiev, Alexander Zielenski, Alexandra Barr, Alexey Ivanov, Alexi Christakis, Alfred Xue, Allison Tam, Ally Bennett, Ally Bennett , Amelia Liu, Amy McDonald Sandjideh, Ananya Kumar, Andre Saraiva, Andrea Vallone, Andrew Chen, Andrew Duberstein, Andrew Gibiansky, Andrew Kondrich, Andrew Tulloch, Andrey Mishchenko, Andy Applebaum, Andy Wang, Angela Baek, Annie Wei, Anting Shen, Antoine Pelisse, Anuj Saharan, Arun Vijayvergiya, Ashley Tyra, Ashvin Nair, Avi Nayak, Avital Oliver, Behrooz Ghorbani, Belinda Truong, Ben Sokolowsky, Beth Hoover, Bo Xu, Boaz Barak, Bohan Zhang, Borys Minaiev, Botao Hao, Bowen Baker, Bowen Cheng, Brandon McKinzie, Brandon Wang, Brian Hsu, Brian Yang, Brian Yu, Brian Zhang, Camillo Lugaresi, Carolina Paz, Carpus Chang, Cary Bassin , Cary Hudson, Casey Chu, Chak Li, Charles Zhao, Charlie Jatt, Charlotte Cole, Chelsea Voss, Chen Shen, Chengxu Zhuang, Chris Colby, Chris Hallacy , Chris Koch, Christina Kaplan, Christina Kim, Colin Reid, Colin Wei, Cristina Scheau, D. Sculley, Damien Deville, Dan Roberts, Dana Palmie, Dane Stuckey, Daniel Levine, David Hu, David Martin, David Robinson, David Sasaki, Davis Wu, Derek Chen, Dibya Bhattacharjee, Dimitris Tsipras, Dinghua Li, DJ Strouse, dmed Medina, Drew Hintz, Eddie Zhang, Edmund Wong, Elaine Ya Le, Eli Yani , Elizabeth Proehl, Emily Sokolova, Enoch Cheung, Eri Schwartz, Eric Mitchell, Eric Ning, Eric Sigler, Eric Wallace, Eugenio Panero, Evan Mays, Evgenii Nikishin, Fan Wang, Fangyuan Li, Filippo Raso, Foivos Tsimpourlas, Fouad Matin, Francis Song, Francis Zhang, Gary Yang, Gene Oden, Giambattista Parascandolo, Gildas Chabot, Grace Kim, Grace Zhao, Greg Brockman, Gregory Valiant, Guillaume Leclerc, Hadi Salman, Haitang Hu, Hannah Sheahan, Hao Sheng, Haoyu Wang, Henrique Ponde de Oliveira Pinto, Henry Aspegren, Heqing Yan, Hessam Bagherinezhad, Hongyu Ren, Hunter Lightman, Hyeonwoo Noh, Ian Kivlichan, Ian Sohl, Ignasi Clavera, Ikai Lan, Ilge Akkaya, Ilya Kostrikov, Irina Kofman, Isa Fulford, Jake Brill, Jakub Pachocki, James Betker, James Lee, James Qin, Jamie Kiros, Jason Ai, Jay Wang, Jean Harb, Jeff Mickey, Jeffrey Han, Jeffrey Wang, Jeremy Chen, Jerry Tworek, Jessica Liang, Jessica Shieh, Ji Lin, Jiahui Yu, Jianfeng Wang, Jie Tang, Jihan Yin, Jing Li, Joanne Jang, Joel Morris, Johannes Ferstad, Johannes Heidecke, John Fishbein, Jon Okun, Jonathan Gordon, Joost Huizinga, Jos Kraaijeveld, Joseph Mo, Josh Lawson , Josh Tobin, Junhua Mao, Kai Chen, Kai Hayashi, Karan Singhal, Karina Nguyen, Katy Shi, Kelly Stirman, Kenji Hata, Kenny Nguyen, Keren Gu-Lemberg, Kevin Gladstone, Kevin King, Kevin Liu, Kevin Lu, Kevin Park, Kevin Stone, Kevin Weil, Kevin Whinnery, Kevin Yu, Kote Mushegiani, Kristen Ying, Kristian Georgiev, Kshitij Gupta, Kyle Kosic, Lama Ahmad, Larry Lv, Lauren Itow, Lauren Yang, Lee Byron, Leo Chen, Leo Liu, Leon Maksin, Leyton Ho, Li Jing, Liang Xiong, Lin Yang, Linden Li, Lorenz Kuhn, Louis Feuvrier, Lu Zhang, Łukasz Kaiser, Mahmoud Eariby, Maja Trębacz, Manas Joglekar, Manoli Liodakis, Manuka Stratta, Mark Chen, Mark Hudnall, Mark Sun, Mark Wang, Martin Li, Marvin Zhang, Mateusz Litwin, Matt Jones, Matt Lim, Max Johnson, Max Schwarzer, Mayank Gupta, Meghan Shah, Mengqing Wang, Mengyuan Yan, Mia Glaese, Michael Bolin, Michael Lampe, Michael Malek, Michael Sharman, Michael Zhang, Michele Wang, Michelle Pokrass, Miguel Oom Temudo de Castro, Mihai Florian, Mike McClay, Mike Trpcic, Miki Habryn, Miles Wang, Ming Chen, Mingxuan Wang, Minnia Feng, Mitchell Gordon, Mo Bavarian, Mostafa Rohaninejad, Nacho Soto, Nakul Khanna, Nat McAleese, Natalie Staudacher, Natan LaFontaine, Neel Ajjarapu, Nick Felt, Nick Turley, Nikil Pancha, Nikita Mikhaylin, Niko Felix, Nikunj Handa, Ning Liu, Nishant Rai, Noah Jorgensen, Noam Brown, Oleg Boiko, Oleg Murk, Olivia Watkins, Olivier Godement, Oona Gleeson, Paul Ashbourne, Pavel Belov, Peter Flockhart, Peter Hoeschele, Peter Zhokhov, Philip Pronin, Phillip Guo, Phoebe Thacker, Prafulla Dhariwal, Prashanth R, Rachel Dias, Rahul Arora, Rajkumar Samuel, Rasmus Rygaard, Ravi Teja Mullapudi, Raymond Li, Raz Gaon, Reah Miyara, Reiichiro Nakano, Reimar Leike, Rennie Song, Rhythm Garg, RJ Marsan, Robert Xiong, Robin Brown, Roman Tsiupa, Rui Shu, Ruslan Nigmatullin, Saachi Jain, Saagar Patel, Sam Altman, Sam Toizer, Sam Toyer, Samir Ahmed, Samuel Miserendino, Samuel Wolrich , Sandhini Agarwal, Santiago Hernández, Sarah Dong, Savannah Heon, Scott Ethersmith, Scott Mayer McKinney, Sean Fitzgerald, Sever Banesiu, Shamez Hemani, Shengjia Zhao, Shengli Hu, Shibani Santurkar, Shreyas Krishnaswamy, Shuchao Bi, Shunyu Yao, Shuyuan Zhang, Simón Posada Fishman, Spencer Papay, Spug Golden, Srinivas Narayanan, Stanley Hsieh, Stephen Logsdon, Sundeep Tirumalareddy, Tal Stramer, Tao Wang, Tao Xin, Taylor Gordon, Tejal Patwardhan, Thibault Sottiaux, Tina Sriskandarajah, Tony Casparro, Tony Zhao, Trevor Creech, Uzair Navid Iftikhar, Valerie Qi, Vineet Kosaraju, Vishal Kuo, Vitchyr Pong, Vivek Verma, Vlad Petrov, Wenda Zhou, Wenlei Xie, Wenting Zhan, Will DePue, Will Ellsworth, William Sheu, Wyatt Thompson, Yaming Lin, Yann Dubois, Yaodong Yu, Yara Khakbaz, Yash Patil, Yifan Wu, Yilong Qin, Yining Chen, Yirui Zhang, Yo Shavit , Young Cha, Yunyun Wang, Yushi Wang, Zack Sultan, Zehao Dou, Zewei Chu, Zheng Shao, Zhigang Wang, Zhishuai Zhang, Zihao Zhang