16 Aprili 2025

Introducing OpenAI o3 and o4-mini

Inapakia…

Sasisho mnamo tarehe 10 Juni, 2025: OpenAI o3‑pro sasa inapatikana kwa watumiaji wa Pro katika ChatGPT, pamoja na katika API yetu. Kama OpenAI o1‑pro, o3‑pro ni toleo la muundo wetu wa hali ya juu zaidi, OpenAI o3, ulioundwa kufikiria kwa muda mrefu na kutoa majibu ya kuaminika zaidi. Maelezo kamili yanapatikana katika maelezo ya kutolewa⁠(fungua katika dirisha jipya).

Leo, tunatoa OpenAI o3 na o4-mini, ya hivi karibuni katika mfululizo wetu wa miundo iliyofunzwa ili kuwaza kwa muda mrefu kabla ya kujibu. Hii ndiyo miundo mahiri zaidi ambayo tumetoa hadi sasa, ikionyesha mabadiliko makubwa katika uwezo wa ChatGPT kwa kila mtu, kuanzia watumiaji wadadisi hadi watafiti wa hali ya juu. Kwa mara ya kwanza, mifano yetu ya uwazaji inaweza kwa uhuru kutumia na kuchanganya kila zana ndani ya ChatGPT—hii inajumuisha kutafuta kwenye wavuti, kuchanganua faili zilizopakiwa na data nyingine kwa kutumia Python, kuwaza kwa kina kuhusu viingizo vya kuona na hata kuzalisha picha. Kwa umuhimu, miundo hii imefunzwa kuwaza kuhusu wakati na jinsi ya kutumia zana ili kutoa majibu ya kina na ya kufikiria katika mfumo sahihi ya utoaji, kawaida chini ya dakika moja, kutatua matatizo magumu zaidi. Hii inawaruhusu kushughulikia maswali yenye vipengele vingi kwa ufanisi zaidi, hatua kuelekea ChatGPT yenye uwezo zaidi ambayo inaweza kutekeleza shughuli kwa uhuru kwa niaba yako. Nguvu ya pamoja ya uwazaji wa kisasa na ufikiaji kamili wa zana hutafsiriwa kuwa utendaji thabiti zaidi katika vigezo vya kitaaluma na shughuli za ulimwengu halisi na kuweka kiwango kipya katika maarifa na manufaa.

Kipi kimebadilika

OpenAI o3 ni muundo wetu wenye nguvu zaidi wa hoja ambao unasukuma mipaka katika usimbaji, hesabu, sayansi, mtazamo wa kuona, na zaidi. Inaweka kiwango kipya cha SOTA kwenye viwango vya majaribio ikiwa ni pamoja na Codeforces, SWE-bench (bila kujenga scaffold maalum ya muundo maalum), na MMMU. Ni bora kwa maswali tata yanayohitaji uchambuzi wa pande nyingi na majibu ambayo hayawezi kuwa dhahiri mara moja. Inafanya vizuri sana katika shughuli za kuona kama kuchambua picha, chati, na michoro. Katika tathmini za wataalam wa nje, o3 hufanya makosa makubwa kwa asilimia 20 kidogo kuliko OpenAI o1 kwenye shughuli ngumu, za ulimwengu wa kweli - hasa ikifaulu katika maeneo kama programu, Business/ushauri, na mawazo ya ubunifu. Wajaribu wa awali walionyesha ukali wake wa uchambuzi kama mshirika wa mawazo na wakasisitiza uwezo wake wa kuzalisha na kutathmini kwa kina dhana mpya - hasa katika muktadha wa biolojia, hesabu, na uhandisi.

OpenAI o4-mini ni muundo mdogo ulioboreshwa kwa ajili ya hoja za haraka na za gharama nafuu—hufanikisha utendakazi wa ajabu kwa ukubwa na gharama yake, hasa katika hesabu, usimbaji na shughuli za kuona. Ni muundo unaofanya vizuri zaidi uliowekewa alama kwenye AIME 2024 na 2025. Ingawa ufikiaji wa kompyuta hupunguza kwa kiasi kikubwa ugumu wa mtihani wa AIME, pia tuliona kuwa ni muhimu kwamba o4-mini inapata 99.5% pass@1 (100% consensus@8) kwenye AIME 2025 inapopewa ufikiaji wa mkalimani wa Python. Ingawa matokeo haya hayapaswi kulinganishwa na utendaji wa miundo bila ufikiaji wa zana, ni mfano mmoja wa jinsi o4-mini inavyotumia zana zinazopatikana kwa ufanisi; o3 inaonyesha maboresho sawa kwenye AIME 2025 kutokana na matumizi ya zana (98.4% pass@1, 100% consensus@8).

Katika tathmini za kitaalam, o4-mini pia inashinda mtangulizi wake, o3‑mini, kwenye shughuli zisizo za STEM na pia katika kikoa kama sayansi ya data. Shukrani kwa ufanisi wake, o4-mini hutoa usaidizi wa vikomo vya matumizi vya juu zaidi kuliko o3, na kuifanya kuwa chaguo thabiti la kiwango cha juu cha matumizi na throughput kwa maswali yanayofaidi kutokana na hoja. Wataalam wa nje walikadiria miundo yote miwili kuwa inaonyesha ufuatiliaji wa maagizo ulioboreshwa na majibu muhimu zaidi, yanayoweza kuthibitishwa kuliko watangulizi wao, kutokana na akili iliyoboreshwa na ujumuishaji wa vyanzo vya wavuti. Ikilinganishwa na matoleo ya awali ya miundo yetu ya hoja, miundo hii miwili inapaswa pia kuhisi ya asili zaidi na ya mazungumzo, hasa inaporejelea kumbukumbu na mazungumzo ya zamani ili kufanya majibu kuwa ya kibinafsi zaidi na muhimu.

Multimodal

Uandishi wa msimbo

Tathmini zote za SWE-bench hutumia seti ndogo ya n=477 za shughuli zilizothibitishwa ambazo zimehakikiwa kwenye miundombinu yetu ya ndani.

Kufuata maagizo na matumizi ya zana ya kiwakala

Miundo yote inatathminiwa katika mipangilio ya juu ya 'jitihada za kufikiri'—sawa na vibadala kama vile 'o4-mini-high' katika 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.

Kufikiria kwa kutumia picha

Kwa mara ya kwanza, miundo hii inaweza kuunganisha picha moja kwa moja katika mlolongo wao wa mawazo. Hawaoni tu picha—wanawaza nayo. Hii inafungua darasa jipya la utatuzi wa matatizo linalochanganya mawazo ya kuona na maandishi, linaloonyeshwa katika utendaji wao wa hali ya juu katika viwango vya alama za kimataifa vya multimodal.

Watu wanaweza kupakia picha ya ubao mweupe, mchoro wa kitabu cha kiada, au mchoro uliochorwa kwa mkono, na muundo unaweza kuifasiri—hata kama picha hiyo ni ya ukungu, imegeuzwa au ina ubora wa chini. Kwa kutumia zana, miundo inaweza kudhibiti picha papo hapo—kuzizungusha, kuzikaribisha, au kuziunda upya kama sehemu ya mchakato wao wa kufikiria.

Miundo hii inatoa usahihi wa hali ya juu katika shughuli za mtazamo wa kuona, ikiwezesha kutatua maswali ambayo hapo awali hayakuweza kufikiwa. Tazama blogi ya utafiti wa uwazaji wa kuona⁠ ili kujifunza zaidi.

Kuelekea matumizi ya zana za kiwakala

OpenAI o3 na o4-mini zina ufikiaji kamili wa zana ndani ya ChatGPT, pamoja na zana zako maalum kupitia kupiga kazi katika API. Miundo hii imefunzwa Waza jinsi ya kutatua matatizo, kuchagua wakati na jinsi ya kutumia zana kutoa majibu ya kina na ya kufikiria katika miundo sahihi ya utoaji haraka—kwa kawaida chini ya dakika moja.

Kwa mfano, mtumiaji anaweza kuuliza: "Je, matumizi ya nishati ya majira ya joto huko California yatalinganishwaje na mwaka jana?" Muundo unaweza kutafuta wavuti kwa data ya matumizi ya umma, kuandika msimbo wa Python kuunda utabiri, kuzalisha grafu au picha, na kueleza mambo muhimu nyuma ya utabiri, huku ukiunganisha simu nyingi za zana. Hoja inaruhusu miundo kuitikia na kubadilika inapohitajika kwa habari inayokutana nayo. Kwa mfano, wanaweza kutafuta wavuti mara nyingi kwa usaidizi wa watoa huduma za utafutaji, kuangalia matokeo na kujaribu utafutaji mpya ikiwa wanahitaji maelezo zaidi.

Mbinu hii inayobadilika na ya kimkakati inaruhusu miundo kushughulikia shughuli zinazohitaji ufikiaji wa taarifa za kisasa zaidi ya maarifa yaliyojengewa ndani ya muundo, hoja za kina, usanisi na utoaji wa matokeo katika njia mbalimbali.

Mifano yote ilikamilishwa na 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)\ ).

Iliyowazwa kuhusu ujenzi wa polinomial kwa sekunde 55

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:

Alifikiria kwa 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}

OpenAI o3 hupata jibu kwa usahihi bila kutumia utafutaji, ilhali o1 inashindwa kutoa jibu sahihi.

Kuendeleza hoja za gharama nafuu

Gharama dhidi ya utendaji: o3‑mini na o4-mini

Gharama dhidi ya utendaji: o1 na o3

OpenAI o3 na o4-mini ni miundo yenye akili zaidi ambayo tumewahi kutoa, na pia mara nyingi huwa na ufanisi zaidi kuliko watangulizi wao, OpenAI o1 na o3‑mini. Kwa mfano, kwenye shindano la hesabu la AIME la 2025, mpaka wa utendaji wa gharama wa o3 unaboreka zaidi ya o1 na vile vile, mpaka wa o4-mini unaboreka zaidi ya o3‑mini. Kwa ujumla, tunatarajia kwamba kwa matumizi mengi ya ulimwengu halisi, o3 na o4-mini pia zitakuwa na akili zaidi na za bei nafuu kuliko o1 na o3‑mini, mtawalia.

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: hoja za kimaendeleo kwenye terminali

Pia tunashiriki jaribio jipya: Codex CLI, wakala mwepesi wa usimbaji ambao unaweza kuendesha kutoka kwenye terminali yako. Inafanya kazi moja kwa moja kwenye kompyuta yako na imeundwa ili kuongeza uwezo wa kufikiri wa miundo kama o3 na o4-mini, huku ikitarajiwa kutoa usaidizi kwa miundo mingine ya API kama GPT‑4.1⁠.

Unaweza kupata manufaa ya uamuzi wa multimodal kutoka kwa mstari wa amri kwa kupitisha picha za skrini au michoro ya uaminifu wa chini kwa muundo, pamoja na ufikiaji wa msimbo wako ndani ya nchi. Tuwaza kama kiolesura cha msingi cha kuunganisha miundo yetu na watumiaji pamoja na kompyuta zao. Codex CLI ni chanzo wazi kabisa kwenye github.com/openai/codex⁠(fungua katika dirisha jipya) Leo.

Pamoja na hayo, tunazindua mpango wa $1 milioni wa usaidizi wa miradi inayotumia Codex CLI na miundo ya OpenAI. Tutatathmini na kukubali programu za ruzuku kwa nyongeza ya $25,000 USD katika mfumo wa mikopo ya API. Mapendekezo yanaweza kuwasilisha hapa.

Ufikiaji

Watumiaji wa ChatGPT Plus, Pro, na Team wataona o3, o4-mini, na o4-mini-high katika kichagua muundo kuanzia leo, wakibadilisha o1, o3‑mini, na o3‑mini‑high. Watumiaji wa ChatGPT Enterprise na Edu watapata ufikiaji katika wiki moja. Watumiaji wa Free wanaweza kujaribu o4-mini kwa kuchagua 'waza' katika mtunzi kabla ya kuwasilisha swali lao. Vikomo vya viwango katika mipango yote havijabadilika kutoka kwa seti ya awali ya miundo.

Tunatarajia kuachilia OpenAI o3‑pro baada ya wiki chache na usaidizi kamili wa zana. Kwa sasa, watumiaji wa Pro bado wanaweza kufikia o1‑pro.

O3 na o4-mini zinapatikana pia kwa wasanidi programu leo kupitia API ya Kukamilisha Chati na API ya Majibu (baadhi ya wasanidi programu watahitaji thibitisha mashirika yao⁠(fungua katika dirisha jipya) ili kufikia miundo hii). API ya Majibu inasaidia muhtasari wa hoja, uwezo wa kuhifadhi tokeni za hoja karibu na miito ya kazi kwa utendaji bora, na hivi karibuni itasaidia zana zilizojengewa ndani kama vile utafutaji wa wavuti, utafutaji wa faili, na Kifasiri cha Msimbo ndani ya hoja za muundo. Ili kuanza, gundua hati zetu⁠(fungua katika dirisha jipya) na endelea kusubiri masasisho zaidi.

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(fungua katika dirisha jipya) 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.

Rudia matangazo ya moja kwa moja

Mwandishi

OpenAI

Tanbihi

* Nambari za tathmini za tau-benchi ni wastani wa mikimbio 5 ili kupunguza tofauti, na zinaendeshwa bila zana maalum au maombi yoyote. Tuligundua kuwa utoaji wa rejareja wa tau-benchi unakabiliwa zaidi na makosa ya muundo wa mtumiaji. Nambari za upau uliotiwa kivuli zinaendeshwa na GPT-4.1 kama muundo wa mtumiaji kwa sababu ni bora zaidi kuliko GPT-4o katika kufuata maagizo.

* SWE-bench hutumia urefu wa juu wa muktadha wa 256k, ambayo inaboresha kiwango cha utatuzi cha o4-mini kwa takriban 3% na inaathiri kiwango cha utatuzi cha o3 kwa <1%. Pia tunatoa nje sampuli 23 ambazo haziwezi kuendeshwa kwenye miundombinu yetu ya ndani.

* Wakati wa kuwezesha kuvinjari, muundo unaweza wakati mwingine kupata majibu sahihi mtandaoni, kwa mfano, kwa kusoma machapisho ya blogi yenye matatizo ya sampuli kutoka kwa seti ya data. Tunapunguza wasiwasi kuhusu udanganyifu wa muundo wakati wa kuvinjari kwa kutumia mikakati miwili:

Vikoa vilivyofungiwa ambavyo tumegundua muundo una danganya kutoka hapo awali.
Ilitumia muundo wa uwazaji kama mfuatiliaji kuchunguza tokeni zote katika kila jaribio, ili kubaini tabia ya kutiliwa shaka. Tabia ya kutiliwa shaka inafafanuliwa kama "ukurasa, faili, au kipande ambacho lengo kuu ni kutoa jibu sahihi kwa swali hili maalum—kwa mfano, ufunguo rasmi wa alama, muhtasari wa “suluhisho” uliovuja, au majadiliano yanayonukuu jibu lililokamilika neno kwa neno." Tabia isiyo na madhara inafafanuliwa kama "Rasilimali yoyote yenye mamlaka ambayo binadamu mwenye bidii anaweza kushauriana (nyaraka, miongozo, karatasi za kitaaluma, makala zinazoheshimika) hata ikiwa kwa bahati ina jibu sahihi kwa bahati." Jaribio lolote ambalo mfuatiliaji aliona kuwa lina tabia ya kutiliwa shaka linahesabiwa kuwa si sahihi. Sampuli nyingi zilizoshindwa na ukaguzi huu zilikuwa matatizo ambayo suluhisho sahihi lilipatikana kwenye vyanzo vingi vya mtandao visivyo na uhusiano na HLE.

Hatimaye, tathmini zetu za kuvinjari huenda zisiweze kuzalishwa kikamilifu katika API ya OpenAI kutokana na tofauti ya viambajengo vya injini za utafutaji kati ya ChatGPT na API ya OpenAI. Matokeo haya yanakusudiwa kuwakilisha uzoefu wa mtumiaji wa ChatGPT, lakini kulingana na mahitaji, usanidi wa utafutaji unaweza kubadilika baada ya muda.

Wachangiaji

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