Abril 16, 2025

Introducing OpenAI o3 and o4-mini

Naglo-load…

Update on June 10, 2025: OpenAI o3‑pro is now available to Pro users in ChatGPT, as well as in our API. Like OpenAI o1‑pro, o3‑pro is a version of our most intelligent model, OpenAI o3, designed to think longer and provide the most reliable responses. Full details can be found in our release notes⁠(magbubukas sa bagong window).

Today, we’re releasing OpenAI o3 and o4-mini, the latest in our o-series of models trained to think for longer before responding. These are the smartest models we’ve released to date, representing a step change in ChatGPT's capabilities for everyone from curious users to advanced researchers. For the first time, our reasoning models can agentically use and combine every tool within ChatGPT—this includes searching the web, analyzing uploaded files and other data with Python, reasoning deeply about visual inputs, and even generating images. Critically, these models are trained to reason about when and how to use tools to produce detailed and thoughtful answers in the right output formats, typically in under a minute, to solve more complex problems. This allows them to tackle multi-faceted questions more effectively, a step toward a more agentic ChatGPT that can independently execute tasks on your behalf. The combined power of state-of-the-art reasoning with full tool access translates into significantly stronger performance across academic benchmarks and real-world tasks, setting a new standard in both intelligence and usefulness.

Ano ang nagbago

Ang OpenAI o3 ay ang aming pinakamahusay na modelo ng pagdadahilan na nilalampasan ang hangganan sa coding, matematika, agham, visual na pang-unawa, at iba pa. Nagtatakda ito ng bagong SOTA sa mga benchmark tulad ng Codeforces, SWE-bench (nang hindi bumubuo ng custom na scaffold na partikular sa modelo), at MMMU. Napakainam nito para sa mga kumplikadong query na nangangailangan ng maraming aspeto ng pagsusuri at kung saan hindi kaagad halata ang mga sagot. Lalong mahusay ang peformance nito sa mga visual na gawain tulad ng pagsusuri ng mga larawan, chart, at graphic. Sa mga pagsusuri ng mga panlabas na eksperto, 20 porsiyentong mas kaunti ang mga nagagawang pagkakamali ng o3 kaysa sa OpenAI o1 sa mahihirap na gawain sa totoong mundo—at lalong mahusay ito sa mga larangan tulad ng programming, business/consulting, at malikhaing pagbuo ng ideya. Binigyang-pansin ng mga naunang tester ang analytical rigor nito bilang katuwang sa pag-iisip at binigyang-diin nila ang kakayahan nitong bumuo at kritikal na sumuri ng mga bagong hypothesis—lalo na sa mga konteksto ng biology, matematika, at engineering.

Ang OpenAI o4-mini ay isang mas maliit na modelo na na-optimize para sa mabilis at sulit na pagdadahilan—kahanga-hanga ang performance nito para sa laki at halaga nito, lalo na sa matematika, coding, at mga visual na gawain. Ito ang modelong na-benchmark na may pinakamahusay na performance sa AIME 2024 at 2025. Bagama't lubos na nababawasan ang hirap ng AIME exam dahil sa pagkakaroon ng access sa computer, napansin din namin na nakakuha ang o4-mini ng 99.5% pass@1 (100% consensus@8) sa AIME 2025 nang binigyan ito ng access sa Python interpreter. Bagama't hindi dapat ihambing ang mga resultang ito sa performance ng mga modelong walang access sa tool, isang halimbawa ang mga ito kung gaano kabisa ang o4-mini sa paggamit ng mga available na tool; nagpakita ang o3 ng mga katulad na paghusay sa AIME 2025 mula sa paggamit ng tool (98.4% pass@1, 100% consensus@8).

Sa mga pagsusuri ng eksperto, mas maganda rin ang performance ng o4-mini kaysa sa performance ng nauna ritong o3‑mini sa mga gawaing walang kinalaman sa STEM pati na rin sa mga domain tulad ng data science. Dahil sa kahusayan nito, nasusuportahan ng o4-mini ang higit na mas matataas na limitasyon sa paggamit kaysa sa o3, kaya mahusay itong opsyon na may malaking volume at malaking throughput para sa mga tanong kung saan makakatulong ang pagdadahilan. Ni-rate ng mga panlabas na ekspertong tagasuri ang dalawang modelo bilang nagpapakita ng mas mahusay na pagsunod sa mga tagubilin at mga mas kapaki-pakinabang at napapatunayang sagot kaysa sa mga nauna sa mga ito dahil sa pinahusay na intelligence at pagsasama ng mga web source. Kumpara sa mga nakaraang iteration ng aming mga modelo ng pagdadahilan, mas natural at parang nakikipag-usap din ang dalawang modelong ito, lalo na dahil sumasangguni ang mga ito sa memory at mga nakaraang pag-uusap upang gawing mas naka-personalize at nauugnay ang mga sagot.

Multimodal

Pag-code

Lahat ng mga takbo ng pagsusuri ng SWE-bench ay gumagamit ng nakapirming subset ng n=477 na beripikadong gawain na napatunayan sa aming panloob na imprastruktura.

Pagsunod sa mga tagubilin at paggamit ng mga tool ng ahente

Sinusuri ang lahat ng modelo sa mga setting ng mataas na ‘pagsisikap sa pagdadahilan’—katulad ng mga variant gaya ng ‘o4-mini-high’ sa ChatGPT.

Continuing to scale reinforcement learning

Throughout the development of OpenAI o3, we’ve observed that large-scale reinforcement learning exhibits the same “more compute = better performance” trend observed in GPT‑series pretraining. By retracing the scaling path—this time in RL—we’ve pushed an additional order of magnitude in both training compute and inference-time reasoning, yet still see clear performance gains, validating that the models’ performance continues to improve the more they’re allowed to think. At equal latency and cost with OpenAI o1, o3 delivers higher performance in ChatGPT—and we've validated that if we let it think longer, its performance keeps climbing.

We also trained both models to use tools through reinforcement learning—teaching them not just how to use tools, but to reason about when to use them. Their ability to deploy tools based on desired outcomes makes them more capable in open-ended situations—particularly those involving visual reasoning and multi-step workflows. This improvement is reflected both in academic benchmarks and real-world tasks, as reported by early testers.

Pag-iisip gamit ang mga larawan

Sa kauna-unahang pagkakataon, makakapag-integrate ang mga modelong ito ng mga larawan nang direkta sa chain of thought ng mga ito. Hindi lang nakakakita ang mga ito ng larawan—nag-iisip ang mga ito gamit ito. Ina-unlock nito ang bagong klase ng paglutas ng problema na pinagsasama ang visual at textual na pagdadahilan, na makikita sa pinakamakabagong performance ng mga ito sa mga multimodal na benchmark.

Puwedeng mag-upload ang mga tao ng larawan ng whiteboard, diagram ng textbook, o sketch na iginuhit ng kamay, at puwede itong bigyang-kahulugan ng modelo—kahit na malabo, baligtad, o mababa ang kalidad ng larawan. Sa paggamit ng tool, magagawa ng mga modelo na manipulahin ang mga larawan nang mabilis—ikutin, i-zoom, o baguhin ang mga ito bilang bahagi ng proseso ng pagdadahilan ng mga ito.

Ang mga modelong ito ay naghahatid ng pinakamahusay na katumpakan sa mga gawain sa visual na pang-unawa, na nagbibigay-daan sa mga ito na sagutin ang mga tanong na hindi masagot noon. Tingnan ang blog sa pananaliksik sa visual na pagdadahilan⁠ para matuto pa.

Tungo sa paggamit ng agentic tool

Ang OpenAI o3 at o4-mini ay may ganap na access sa mga tool sa ChatGPT, pati na rin sa mga sarili mong custom na tool sa pamamagitan ng function calling sa API. Sinanay ang mga modelong ito upang magdahilan kung paano lutasin ang mga problema, at piliin kung kailan at paano gamitin ang mga tool upang makabuo ng mga detalyado at pinag-isipang sagot sa mga tamang format ng output nang mabilis—karaniwang wala pang isang minuto.

Halimbawa, maaaring magtanong ang isang user: “Paano maikukumpara sa nakaraang taon ang paggamit ng enerhiya sa tag-init sa California?“ Puwedeng maghanap ang modelo sa web ng data ng pampublikong utility, magsulat ng Python code para bumuo ng forecast, bumuo ng graph o larawan, at ipaliwanag ang mga pangunahing salik sa likod ng prediksyon, at pagsasama-samahin nito ang maraming tool call. Ang pagdadahilan ay nagbibigay-daan sa mga modelo na tumugon at mag-pivot kung kinakailangan sa impormasyong makikita ng mga ito. Halimbawa, puwedeng maghanap ang mga ito sa web nang maraming beses sa tulong ng mga provider ng paghahanap, tingnan ang mga resulta, at sumubok ng mga bagong paghahanap kung kailangan ng mga ito ng higit pang impormasyon.

Nagbibigay-daan ang flexible at madiskarteng paraang ito sa mga modelo na gawin ang mga gawain na nangangailangan ng access sa updated na impormasyon na higit pa sa built-in na kaalaman ng modelo, pinalawak na pagdadahilan, synthesis, at pagbuo ng output sa iba't ibang modalidad.

Ang lahat ng halimbawa ay nakumpleto gamit ang 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)\ ).

Nagdahilan tungkol sa paggawa ng polynomial sa loob ng 55 segundo

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:

Nag-isip nang 1m 19s

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}

Tama ang nagiging sagot ng OpenAI o3 nang hindi gumagamit ng paghahanap, samantalang hindi nakakapagbigay ng tamang sagot ang o1.

Pagsusulong ng sulit na pagdadahilan

Gastos kumpara sa performance: o3‑mini at o4-mini

Mga pagsusuri sa OpenAI o3-mini at o4-mini gamit ang AIME

Mga pagsusuri sa OpenAI o3-mini at o4-mini gamit ang GPQA

Gastos kumpara sa performance: o1 at o3

Mga pagsusuri sa OpenAI o3 at o4 gamit ang AIME

Mga pagsusuri sa OpenAI o3 at o4 gamit ang GPQA

Ang OpenAI o3 at o4-mini ay ang pinakamatatalinong modelo na nailabas namin, at madalas ding mas mahusay ang mga ito kaysa sa mga nauna sa mga ito na OpenAI o1 at o3‑mini. Halimbawa, sa kumpetisyon sa matematika ng AIME sa taong 2025, lubos na mas mahusay ang cost-performance frontier para sa o3 kaysa sa o1, at sa parehong paraan, lubos na mas mahusay ang frontier ng o4-mini kaysa sa o3‑mini. Sa pangkalahatan, inaasahan namin na para sa karamihan ng paggamit sa totoong mundo, ang o3 at o4-mini ay magiging mas matalino at mas mura din kaysa sa o1 at o3‑mini, ayon sa pagkakabanggit.

Safety

Each improvement in model capabilities warrants commensurate improvements to safety. For OpenAI o3 and o4-mini, we completely rebuilt our safety training data, adding new refusal prompts in areas such as biological threats (biorisk), malware generation, and jailbreaks. This refreshed data has led o3 and o4-mini to achieve strong performance on our internal refusal benchmarks (e.g., instruction hierarchy⁠, jailbreaks). In addition to strong performance for model refusals, we have also developed system-level mitigations to flag dangerous prompts in frontier risk areas. Similar to our earlier work in image generation⁠, we trained a reasoning LLM monitor which works from human-written and interpretable safety specifications. When applied to biorisk, this monitor successfully flagged ~99% of conversations in our human red‑teaming campaign.

We stress tested both models with our most rigorous safety program to date. In accordance with our updated Preparedness Framework⁠, we evaluated o3 and o4-mini across the three tracked capability areas covered by the Framework: biological and chemical, cybersecurity, and AI self-improvement. Based on the results of these evaluations, we have determined that both o3 and o4‑mini remain below the Framework's "High" threshold in all three categories. We have published the detailed results from these evaluations in the accompanying system card⁠.⁠

Codex CLI: pinaka-advanced na pagdadahilan sa terminal

May ibabahagi rin kaming bagong eksperimento: Codex CLI, isang magaan na coding agent na puwede mong patakbuhin mula sa iyong terminal. Direkta itong gumagana sa iyong computer at idinisenyo upang i-maximize ang mga kakayahan sa pagdadahilan ng mga modelo tulad ng o3 at o4-mini, na may paparating na suporta para sa mga karagdagang modelo ng API tulad ng GPT‑4.1⁠.

Makukuha mo ang mga benepisyo ng multimodal na pagdadahilan mula sa command line sa pamamagitan ng pagpapasa ng mga screenshot o low fidelity na sketch sa modelo, kasabay ng lokal na pag-access sa iyong code. Itinuturing namin ito bilang isang minimal na interface para ikonekta ang aming mga modelo sa mga user at kanilang mga computer. Ang Codex CLI ay ganap na open-source sa github.com/openai/codex⁠(magbubukas sa bagong window) ngayon.

Kasabay nito, maglulunsad kami ng inisyatibang nagkakahalaga ng $1 milyon para suportahan ang mga proyekto gamit ang Codex CLI at mga modelo ng OpenAI. Susuriin at tatanggapin namin ang mga aplikasyon para sa mga grant na tumataas nang $25,000 USD ang halaga sa anyo ng mga API credit. Puwedeng isumite ang mga proposal dito.

Access

Makikita ng mga user ng ChatGPT Plus, Pro, at Team ang o3, o4-mini, at o4-mini-high sa selector ng modelo simula ngayon, na papalit sa o1, o3‑mini, at o3‑mini‑high. Magkakaroon ng access ang mga user ng ChatGPT Enterprise at Edu sa loob ng isang linggo. Puwedeng subukan ng mga user ng Free ang o4-mini sa pamamagitan ng pagpili ng 'Mag-isip' sa composer bago isumite ang kanilang query. Hindi magbabago ang mga limitasyon sa rate sa lahat ng plan mula sa naunang hanay ng mga modelo.

Inaasahan naming i-release ang OpenAI o3‑pro sa loob ng ilang linggo nang may ganap na suporta sa tool. Sa ngayon, maa-access pa rin ng mga user ng Pro ang o1‑pro.

Available din ang parehong o3 at o4-mini sa mga developer ngayon sa pamamagitan ng Chat Completions API at Responses API (kakailanganin ng ilang developer na beripikahin ang kanilang mga organisasyon⁠(magbubukas sa bagong window) upang ma-access ang mga modelong ito). Sinusuportahan ng Responses API ang mga buod ng pagdadahilan, ang kakayahang mapanatili ang mga token ng pagdadahilan sa paligid ng mga function call para sa mas mahusay na performance, at malapit nang suportahan ang mga built-in na tool tulad ng paghahanap sa web, paghahanap ng file, at code interpreter sa loob ng pagdadahilan ng modelo. Para magsimula, galugarin ang aming mga doc⁠⁠(magbubukas sa bagong window) at abangan ang iba pang update.

What's next

Today's updates reflect the direction our models are heading in: we’re converging the specialized reasoning capabilities of the o-series with more of the natural conversational abilities and tool use of the GPT‑series. By unifying these strengths, our future models will support seamless, natural conversations alongside proactive tool use and advanced problem-solving.

Update on July 28, 2025: SWE-Lancer dataset and results updated as of July 17, 2025, available at: https://github.com/openai/preparedness⁠(magbubukas sa bagong window) and in our system cards. This update resolves several issues that were impacting the dollars earned results and removes the requirement for internet connectivity during execution, eliminating a primary source of variability in model performance.

Update on April 16, 2025: results for o3 on Charxiv-r and Mathvista were updated to reflect a system prompt change that wasn't present in the original evaluation.

Replay ng livestream

May-akda

OpenAI

Mga Footnote

* Ang mga numero ng pagsusuri gamit ang tau-bench ay ina-average sa 5 pagpapatakbo upang mabawasan ang pagkakaiba-iba, at pinapatakbo ang mga ito nang walang anumang custom na tool o pag-prompt. Napag-alaman naming ang mga retail na paglulunsad ng tau-bench ay mas madaling magkaroon ng mga error sa modelo ng user. Ang mga numero para sa naka-shade na bar ay pinatakbo gamit ang GPT-4.1 bilang modelo ng user dahil lubos na mas mahusay ito kaysa sa GPT-4o sa pagsunod sa mga tagubilin.

* Gumagamit ang SWE-bench ng 256k max na haba ng konteksto, na pinapahusay ang rate ng paglutas para sa o4-mini nang halos 3% at nakakaapekto sa rate ng paglutas ng o3 nang <1%. Hindi rin namin isinama ang 23 sample na hindi maaaring patakbuhin sa aming panloob na imprastraktura.

** Kapag na-enable ang pag-browse, puwedeng makahanap minsan ang modelo ng mga eksaktong sagot online, halimbawa, sa pamamagitan ng pagbabasa ng mga post sa blog na may mga sample na problema mula sa isang dataset. Binabawasan namin ang mga alalahanin tungkol sa pandaraya ng modelo habang nagba-browse gamit ang dalawang estratehiya:

I-block ang mga domain na naobserbahan naming dating ginamit ng modelo sa pandaraya.
Gumamit ng modelo ng pagdadahilan bilang tagapagmasid upang suriin ang lahat ng token sa bawat pagtatangka, upang matukoy ang kahina-hinalang gawi. Tinutukoy ang kahina-hinalang gawi bilang "isang page, file, o snippet na ang pangunahing layunin ay magbigay ng eksaktong sagot sa partikular na tanong na ito—halimbawa, isang opisyal na key sa pagmamarka, kumalat na buod ng “mga solusyon,” o talakayan na nagsisipi sa natapos na sagot nang verbatim." Tinutukoy ang benign na gawi bilang "Anumang mapagkakatiwalaang resource na maaaring konsultahin ng isang masigasig na tao (dokumentasyon, mga manwal, mga scholarly paper, mga mapagkakatiwalaang artikulo) kahit na nagkataong naglalaman ito ng tamang sagot." Ibinibilang na hindi wasto ang anumang pagtatangka na may paglulunsad na itinuturing ng tagapagmasid na kahina-hinala. Ang karamihan sa mga sample na hindi pumasa sa pagsusuring ito ay ang mga problema na may eksaktong solusyon na available sa maraming source sa internet na walang kaugnayan sa HLE.

Panghuli, ang aming mga pagsusuri sa pag-browse ay maaaring hindi ganap na ma-reproduce sa OpenAI API dahil sa pagkakaiba sa mga backend ng search engine ng ChatGPT at ng OpenAI API. Nilalayong maging kinatawan ang mga resultang ito ng karanasan ng user ng ChatGPT, ngunit depende sa demand, maaaring magbago ang configuration ng paghahanap sa paglipas ng panahon.

Mga Contributor

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