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OpenAI

Abriil 16, 2025

ShaacinBadeecad

Introducing OpenAI o3 and o4-mini

Ku tijaabi ChatGPT(ku furmaa daaqad cusub)
Soo kacaya…

Cusboonaysiin Juun 10, 2025: OpenAI o3‑pro hadda waxa uu u diyaar yahay isticmaaleyaasha Pro ee ChatGPT, iyo sidoo kale API-gayaga. Sida OpenAI o1‑pro, o3‑pro waa nooc ka mid ah nooca noogu garaadka badan, OpenAI o3, oo loo naqshadeeyey inuu sii fikiro muddo dheer oo uu bixiyo jawaabaha ugu kalsoonida badan. Faahfaahin buuxda waxaa laga heli karaa qoraallada sii-deynta⁠(ku furmaa daaqad cusub).

Maanta, waxaan sii deynaynaa OpenAI o3 iyo o4-mini, oo ah kuwii ugu dambeeyay ee taxanaheenna o-series ee noocyada loo tababaray inay muddo dheer fikiraan ka hor inta aysan jawaabin. Kuwani waa noocyada ugu caqliga badan ee aan sii daynay ilaa maanta, waxayna matalayaan isbeddel weyn oo ku yimid awoodaha ChatGPT ee qof kasta laga bilaabo isticmaalayaasha xiisaha leh ilaa cilmi-baarayaasha horumarsan. Markii ugu horreysay, noocyadayada caqliyeyntu waxay si wakiilnimo leh u adeegsan karaan oo isu geyn karaan qalab kasta oo ku jira ChatGPT—tan waxaa ku jira raadinta webka, falanqaynta faylasha la soo geliyey iyo xog kale oo lagu sameeyo Python, si qoto dheer uga caqliyeynta gelinnada muuqaalka, iyo xitaa soo saarida sawirro. Waxa muhiimka ah, noocyadan waxaa loo tababaray inay ka caqliyeeyaan goorta iyo sida loo adeegsado qalabka si loo soo saaro jawaabo faahfaahsan oo feker leh qaababka saxda ah ee wax-soo-saarka, badanaa wax ka yar hal daqiiqo gudaheed, si loo xalliyo dhibaatooyin ka sii adag. Tani waxay u oggolaanaysaa inay si wax ku ool ah uga shaqeeyaan su’aalo dhinacyo badan leh, taas oo ah tallaabo ku wajahan ChatGPT wakiilnimo badan leh oo si madax-bannaan hawlo kuugu fulin kara. Awoodda la isku daray ee caqliyeynta heerka ugu sarreysa oo leh helitaan qalab buuxa waxay isu beddeshaa waxqabad si weyn uga xooggan benchmarks-ka tacliimeed iyo hawlaha dunida dhabta ah, iyadoo dejinaysa halbeeg cusub oo dhinaca garaadka iyo faa’iidada labadaba ah.

Waxa is beddelay

OpenAI o3 waa nooca caqliyeynta ee noogu awoodda badan kaas oo riixaya xadka ugu casriyeysan ee koodh qorista, xisaabta, sayniska, aragga muuqaalka, iyo wax ka badan. Waxay dejisaa SOTA cusub oo ku saabsan benchmarks ay ku jiraan Codeforces, SWE-bench (iyadoo aan la dhisin scaffold u gaar ah nooca), iyo MMMU. Waxay ku habboon tahay weydiimaha adag ee u baahan falanqayn dhinacyo badan leh oo jawaabahoodu aysan isla markiiba caddayn. Waxay si gaar ah ugu xooggan tahay hawlaha muuqaalka sida falanqaynta sawirrada, jaantusyada, iyo garaafyada. Qiimeyn ay sameeyeen khubaro dibadeed, o3 waxay samaysaa 20 boqolkiiba khaladaad waaweyn oo ka yar OpenAI o1 hawlaha adag ee dunida dhabta ah—gaar ahaan waxay kaga fiicnaataa meelaha sida barnaamij-samaynta, ganacsiga/la-talinta, iyo curinta fikrado hal-abuur leh. Tijaabiyeyaashii hore waxay iftiimiyeen adkaanteeda falanqayn ahaan lammaane fikir, waxayna xoojiyeen awooddeeda ay ku soo saarto oo si dhaliil leh u qiimeyso mala-awaallo cusub—gaar ahaan xaaladaha bayoolaji, xisaab, iyo injineernimo.

OpenAI o4-mini waa nooc yar oo loo habeeyey caqliyeyn degdeg ah oo kharash-ku-ool ah—waxay gaartaa waxqabad cajiib ah marka loo eego cabbirkeeda iyo kharashkeeda, gaar ahaan xisaabta, koodh qorista, iyo hawlaha muuqaalka. Waa nooca benchmark ahaan ugu waxqabadka fiican AIME 2024 iyo 2025. Inkastoo helitaanka kombiyuutar uu si macno leh u yareeyo adkaanta imtixaanka AIME, waxaan sidoo kale u aragnay mid mudan in la xuso in o4-mini gaarto 99.5% pass@1 (100% consensus@8) AIME 2025 marka la siiyo helitaanka Python interpreter. Inkastoo natiijooyinkan aan lala barbar dhigi karin waxqabadka noocyada aan helin qalab, waxay yihiin hal tusaale oo muujinaya sida wax ku ool ah ee o4-mini uga faa’iidaysato qalabka la heli karo; o3 waxay muujisaa horumarro la mid ah AIME 2025 marka qalab la adeegsado (98.4% pass@1, 100% consensus@8).

Qiimeynaha khubarada, o4-mini sidoo kale waxay ka waxqabad fiican tahay kii ka horreeyay, o3‑mini, hawlaha aan STEM-ka ahayn iyo sidoo kale meelaha sida data science. Waxtarkeeda awgiis, o4-mini waxay taageertaa xadad isticmaal oo si weyn uga sarreeya o3, taasoo ka dhigaysa ikhtiyaar xooggan oo loogu talagalay mug badan iyo throughput sare su’aalaha ka faa’iidaysta caqliyeynta. Qiimeeyayaal khubaro dibadeed ah ayaa ku qiimeeyay labada noocba inay muujiyaan raacid tilmaamo oo hagaagtay iyo jawaabo ka waxtar badan oo la xaqiijin karo marka loo eego kuwii ka horreeyay, taasoo ay ugu wacan tahay garaad la hagaajiyey iyo ku darista ilo web. Marka loo barbardhigo noocyadii hore ee noocyadayada caqliyeynta, labadan nooc waa inay sidoo kale u muuqdaan kuwo dabiici ah oo wada-sheekeysi leh, gaar ahaan marka ay tixraacayaan xusuusta iyo wada-sheekaysiyadii hore si ay jawaabaha uga dhigaan kuwo shakhsiyaysan oo khuseeya.

Habab badan

Koodh qorista

Dhammaan orodyada qiimeynta SWE-bench waxay adeegsadaan qayb go’an oo ka kooban n=477 hawlood oo la xaqiijiyay kuwaas oo lagu ansixiyay kaabeyaashayada gudaha.

Raacitaanka tilmaamaha iyo adeegsiga qalab ee wakiilnimo leh

Dhammaan noocyada waxaa lagu qiimeeyaa dejimaha sare ee ‘reasoning effort’—oo la mid ah kala-doorashooyinka sida ‘o4-mini-high’ ee ChatGPT.

Sii wadista cabbir-korinta waxbarashadda xoojinta ah

Intii lagu jiray horumarinta OpenAI o3, waxaan aragnay in waxbarashadda xoojinta ah ee baaxadda weyn ay muujiso isla jihada “compute badan = waxqabad fiican” ee lagu arkay pretraining-ka taxanaha GPT. Annagoo dib u raacnay jidka scaling-ka—markan RL—waxaan kordhinay hal amar oo dheeraad ah labadaba training compute iyo inference-time reasoning, haddana wali waxaan aragnaa koror waxqabad oo cad, taasoo xaqiijinaysa in waxqabadka noocyadu sii hagaagayo inta badan ee loo oggolaado inay fikiraan. Marka la simo latency iyo kharashka OpenAI o1, o3 waxay keentaa waxqabad sare gudaha ChatGPT—waxaanan xaqiijinay in haddii aan u oggolaano inay sii fikirto muddo dheer, waxqabadkeedu sii kacayo.

Waxaan sidoo kale labada noocba ku tababarnay inay qalab adeegsadaan iyagoo adeegsanaya waxbarashadda xoojinta ah—annagoo barayna kaliya sida loo adeegsado qalabka, balse inay ka caqliyeeyaan goorta loo adeegsanayo. Awooddooda ay qalabka ugu hawlgeliyaan iyadoo lagu salaynayo natiijooyinka la rabo waxay ka dhigaysaa kuwo ka awood badan xaaladaha furan—gaar ahaan kuwa ku lug leh caqliyeyn muuqaal ah iyo socod-shaqooyin tallaabooyin badan leh. Horumarkan wuxuu ka muuqdaa labadaba benchmarks-ka tacliimeed iyo hawlaha dunida dhabta ah, sida ay sheegeen tijaabiyeyaashii hore.

Ku fikirista sawirro

ChatGPT thinking with images

Markii ugu horreysay, noocyadani waxay si toos ah ugu dhex dari karaan sawirro silsiladdooda fikirka. Kaliya sawir ma arkaan—way ku fikiraan. Tani waxay furaysaa fasal cusub oo xalinta dhibaatooyinka ah oo isku dara caqliyeyn muuqaal iyo mid qoraal, taasoo ka muuqata waxqabadkooda heerka ugu sarreeya ee benchmarks-ka multimodal.

Dadka waxay soo gelin karaan sawir sabuurad cad ah, jaantus buug waxbarasho, ama sawir gacmeed, noocuna wuu fasiri karaa—xitaa haddii sawirku qalloocan yahay, rogman yahay, ama tayadiisu hooseyso. Adeegsiga qalabka awgiis, noocyadu waxay sawirrada u maamuli karaan si toos ah—rogid, weyneyn, ama beddelid iyagoo qayb ka ah habkooda caqliyeynta.

Noocyadani waxay bixiyaan saxnaanta ugu fiican ee fasalkooda ee hawlaha aragga muuqaalka, taasoo u suuragelinaysa inay xalliyaan su’aalo hore uga baxsanaa. Eeg blog-ga cilmi-baarista caqliyeynta muuqaalka⁠ si aad wax badan u ogaato.

Ku wajahan adeegsiga qalab ee wakiilnimo leh

OpenAI o3 iyo o4-mini waxay helitaan buuxa u leeyihiin qalabka gudaha ChatGPT, iyo sidoo kale qalabkaaga gaarka ah ee caadiga ah iyada oo loo marayo xusida function-ka ee API-ga. Noocyadan waxaa loo tababaray inay ka caqliyeeyaan sida dhibaatooyinka loo xalliyo, iyagoo dooranaya goorta iyo sida loo adeegsado qalabka si ay si degdeg ah u soo saaraan jawaabo faahfaahsan oo feker leh qaababka saxda ah ee wax-soo-saarka—badanaa wax ka yar hal daqiiqo.

Tusaale ahaan, isticmaale ayaa weydiin kara: “Isticmaalka tamarta xagaaga ee California sidee buu ula barbar dhigi doonaa kii sannadkii hore?” Noocu wuxuu ka baari karaa webka xogta utility-ga dadweynaha, wuxuu qori karaa Python code si uu u dhiso saadaal, wuxuu abuuri karaa garaaf ama sawir, wuxuuna sharxi karaa qodobbada muhiimka ah ee ka dambeeya saadaasha, isaga oo isku xira wicitaanno qalab oo badan. Caqliyeyntu waxay u oggolaanaysaa noocyada inay uga falceliyaan oo ay u leexdaan hadba sida loogu baahan yahay xogta ay la kulmaan. Tusaale ahaan, waxay marar badan ka baadhi karaan webka iyagoo kaashanaya bixiyeyaasha raadinta, eegaya natiijooyinka, oo tijaabinaya raadisyo cusub haddii ay u baahan yihiin macluumaad dheeraad ah.

Habkan dabacsan ee istiraatiijiga ah wuxuu u oggolaanayaa noocyada inay qabtaan hawlo u baahan helitaanka xog cusub oo ka baxsan aqoonta ku dhex dhisan nooca, caqliyeyn dheer, isu-geyn, iyo soo saarid wax-soo-saar oo dhex marta habab kala duwan.

Dhammaan tusaalooyinka waxaa lagu dhammaystiray 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)\ ).

Wuxuu ka caqliyeeyey dhismaha polynomial-ka 55 ilbiriqsi
  • 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).

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Done

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

D0 ⁣(x, a)=2,  D1 ⁣(x, a)=x,  Dn ⁣(x, a)=x Dn−1 ⁣(x, a)−a Dn−2 ⁣(x, a)(n≥2),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

Dn ⁣(u+au, a)=un+anun.D_n\!\left(u + \frac{a}{u},\, a\right) = u^{n} + \frac{a^{n}}{u^{n}}.

1. Choosing the polynomial

Take a=1a = 1 and n=19n=19.
Because nn is odd, D19 ⁣(x, 1)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 xx is x=(−1)n−12 n=−19,x = (-1)^{\frac{n-1}{2}}\,n = -19, so all requested normalisations are satisfied.

Writing it out gives

p(x)=D19 ⁣(x,1)=x19−19x17+152x15−665x13+1729x11−2717x9+2508x7−1254x5+285x3−19x . \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  =  { p(x)=p(y) }fX \;=\; \bigl\{\, p(x) = p(y) \,\bigr\}

For Dickson polynomials one has the classical factorisation

Dn ⁣(x,1)  −  Dn ⁣(y,1)  =  (x−y) ∏j=1n−12(x2+y2−2cos⁡ ⁣(2πjn) xy−4sin⁡2 ⁣(πjn)),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 C\mathbb{C} (see, e.g., Lidl–Niederreiter, Finite Fields, §3.6).
For n=19n=19 this gives

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

Hence X⊂P1×P1X \subset \mathbb{P}^{1} \times \mathbb{P}^{1} has 1010 irreducible components, at least three of them and not all linear, as required.

3. Evaluating p(19)p(19)

Using the recurrence (or the identity Dn ⁣(x, 1)  =  2 Tn ⁣(x2)D_n\!\left(x,\,1\right) \;=\; 2\,T_n\!\bigl(\tfrac{x}{2}\bigr) with Chebyshev TnT_n one obtains

p(19)  =  D19 ⁣(19, 1)  =  1 876 572 071 974 094 803 391 179.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

p(19)  =  1,876,572,071,974,094,803,391,179.\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:

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)\ ).

Wuxuu fikiray 1d 19ilb

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+a17​x17+…+a1​x.

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+a17​x17+…+a3​x3−19x 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 −19-19 whose zero-locus relation { p(x)=p(y) } in P1×P1\{\,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)p(19).

1. The polynomial

Consider

p(x)  =  x3 (x2−1)8  −  19x.p(x) \;=\; x^{3}\,\bigl(x^{2}-1\bigr)^{8} \;-\; 19x.

We can expand

x3 (x2−1)8  =  x19  −  8x17  +  28x15  −  56x13  +  70x11  −  56x9  +  28x7  −  8x5  +  x3.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)  =  x19  −  8x17  +  28x15  −  56x13  +  70x11  −  56x9  +  28x7  −  8x5  +  x3  −  19x.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 x19x^{19}with coefficient 11.

Odd polynomial:
All exponents are odd (19, 17, …, 3, 1),\bigl(19,\,17,\,\ldots,\,3,\,1\bigr), and the constant term is 00. Indeed,

p(−x)  =  − p(x),p(-x) \;=\; -\,p(x),

which makes pp an odd polynomial.

Real coefficients:
All coefficients are real integers.

Linear coefficient −19-19:
The coefficient of xx is clearly −19-19.

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

X  =  { (x,y):p(x)=p(y) }  ⊂  P1×P1X \;=\; \bigl\{\, (x,y) : p(x)=p(y) \,\bigr\} \;\subset\; \mathbb{P}^{1}\times\mathbb{P}^{1}

factors in part because p(x)p(x) has (for instance) the repeated factors (x2−1)8\bigl(x^{2}-1\bigr)^{8} in the big piece x3 (x2−1)8x^{3}\,\bigl(x^{2}-1\bigr)^{8}. One sees that { p(x)=p(y) }\{\,p(x)=p(y)\,\} contains (at least) the divisor $${x=y\}$$ plus further factors coming from the roots x=0  or  x=±1,x = 0 \;\text{or}\; x = \pm 1, ensuring multiple irreducible components (and not all of them are hyperplanes).

2. Computing p(19)p(19)

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

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

Hence the required value is

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

OpenAI o3 waxay si sax ah u heshaa jawaabta iyada oo aan adeegsan raadis, halka o1 uu ku guuldareysto inuu bixiyo jawaab sax ah.

Horumarinta caqliyeyn kharash-ku-ool ah

Kharash vs waxqabad: o3‑mini iyo o4-mini
Qiimeynaha AIME ee OpenAI o3-mini iyo o4-mini
Qiimeynaha GPQA ee OpenAI o3-mini iyo o4-mini
Kharash vs waxqabad: o1 iyo o3
Qiimeynaha AIME ee OpenAI o3 iyo o4
Qiimeynaha GPQA ee OpenAI o3 iyo o4

OpenAI o3 iyo o4-mini waa noocyada ugu garaadka badan ee aan abid sii deynay, waxayna sidoo kale inta badan ka waxtar badan yihiin kuwii ka horreeyay, OpenAI o1 iyo o3‑mini. Tusaale ahaan, tartanka xisaabta AIME 2025, xadka kharash-waxqabadka ee o3 wuxuu si toos ah uga fiican yahay o1, si la mid ahna, xadka o4-mini wuxuu si toos ah uga fiican yahay o3‑mini. Guud ahaan, waxaan filaynaa in inta badan isticmaalka dunida dhabta ah, o3 iyo o4-mini ay sidoo kale ahaan doonaan kuwo ka caqli badan kana jaban o1 iyo o3‑mini, siday u kala horreeyaan.

Badbaado

Horumar kasta oo ku yimaada awoodaha nooca wuxuu u baahan yahay horumar la jaanqaadaya oo badbaado ah. OpenAI o3 iyo o4-mini, waxaan si buuxda dib ugu dhisnay xogtayada tababarka badbaadada, annagoo ku darnay weydiinno diidmo cusub meelo ay ka mid yihiin hanjabaadaha bayoolojiyeed (biorisk), samaynta malware, iyo jailbreaks. Xogtan la cusboonaysiiyay waxay keentay in o3 iyo o4-mini ay gaaraan waxqabad xooggan oo ku saabsan refusal benchmarks-keenna gudaha (tusaale, instruction hierarchy⁠, jailbreaks). Marka lagu daro waxqabadka xooggan ee diidmooyinka nooca, waxaan sidoo kale samaynay yareyno heer-sistam ah si loo calaamadiyo weydiimaha khatarta leh ee meelaha khatarta afka hore. Si la mid ah shaqadeennii hore ee soo saarida sawirka⁠, waxaan tababarnay kormeere reasoning LLM ah oo ka shaqeeya qeexitaanno badbaado oo dadku qoreen oo la fasiri karo. Marka lagu dabaqo biorisk, kormeerahan wuxuu si guul leh u calaamadeeyay ku dhowaad 99% wada-sheekaysiyada ololaheenna human red‑teaming.

Waxaan labada noocba ku samaynay stress test barnaamijkeenna badbaado ee ugu adag ilaa maanta. Iyadoo la raacayo Qaab-dhismeedka Diyaar-garowga⁠ ee la cusboonaysiiyay, waxaan ku qiimeynay o3 iyo o4-mini saddexda meelood ee awoodaha la daba-galo ee uu daboolayo Qaab-dhismeedku: bayooloji iyo kiimiko, amniga internetka, iyo is-horumarinta AI. Iyadoo lagu salaynayo natiijooyinka qiimeyntan, waxaan go’aaminay in labada o3 iyo o4‑mini ay weli ka hooseeyaan xadka "High" ee Qaab-dhismeedka saddexda qayboodba. Waxaan daabacnay natiijooyinka faahfaahsan ee qiimeyntan gudaha kaarka siistamka⁠.⁠

Codex CLI: caqliyeyn ugu casriyeysan oo ku jirta terminal-ka

Waxaan sidoo kale wadaagaynaa tijaabo cusub: Codex CLI, wakiil coding oo fudud oo aad ka socodsiin karto terminal-kaaga. Wuxuu si toos ah uga shaqeeyaa kombiyuutarkaaga waxaana loo naqshadeeyey inuu kordhiyo awoodaha caqliyeynta ee noocyada sida o3 iyo o4-mini, iyadoo ay soo socoto taageero noocyo API dheeraad ah sida GPT‑4.1⁠.

Waxaad ka heli kartaa faa’iidooyinka caqliyeynta multimodal khadka amarka adigoo u gudbinaya screenshots ama sawirro tayadoodu hooseyso nooca, iyadoo lagu daray helitaanka koodhkaaga gudaha deegaankaaga. Waxaan u aragnaa inay tahay interface yar oo lagu xiro noocyadeenna isticmaaleyaasha iyo kombiyuutarradooda. Codex CLI maanta si buuxda ayuu u yahay open-source gudaha github.com/openai/codex⁠(ku furmaa daaqad cusub).

Intaa waxaa dheer, waxaan bilaabaynaa hindise dhan $1 million si loo taageero mashaariicda adeegsada Codex CLI iyo noocyada OpenAI. Waxaan qiimeyn doonnaa oo aqbali doonnaa codsiyada deeqaha oo dhan $25,000 USD ah oo ah API credits. Soo-jeedimaha waxaa lagu gudbin karaa halkan.

Helitaan

Isticmaaleyaasha ChatGPT Plus, Pro, iyo Team waxay bilaabidda maanta ku arki doonaan o3, o4-mini, iyo o4-mini-high xulaha nooca, iyagoo beddelaya o1, o3‑mini, iyo o3‑mini‑high. Isticmaaleyaasha ChatGPT Enterprise iyo Edu waxay heli doonaan toddobaad gudihiis. Isticmaaleyaasha bilaashka ah waxay tijaabin karaan o4-mini iyagoo dooranaya 'Think' gudaha composer-ka ka hor inta aysan dirin weydiintooda. Xaddidaadaha heerka ee dhammaan qorshayaasha waxba iskama beddelin marka loo eego kooxdii noocyadii hore.

Waxaan filaynaa inaan sii deyno OpenAI o3‑pro dhowr toddobaad gudahood iyadoo leh taageero qalab oo buuxda. Hadda, isticmaaleyaasha Pro wali way heli karaan o1‑pro.

Labada o3 iyo o4-mini sidoo kale maanta waxay u heli karaan horumariyayaashu iyada oo loo marayo API-ga dhammeystirka wada-sheekeysiga iyo Responses API (horumariyeyaasha qaar waxay u baahan doonaan inay xaqiijiyaan ururradooda⁠(ku furmaa daaqad cusub) si ay u helaan noocyadan). Responses API waxay taageertaa soo-koobyo caqliyeyn, awoodda lagu keydiyo reasoning tokens agagaarka function calls si waxqabad fiican loo helo, waxayna dhawaan taageeri doontaa qalab dhisan sida web search, file search, iyo code interpreter gudaha caqliyeynta nooca. Si aad u bilowdo, baadh docs-keenna⁠⁠(ku furmaa daaqad cusub) oo la soco cusboonaysiino dheeraad ah.

Maxaa xiga

Cusboonaysiinta maanta waxay ka tarjumayaan jihada ay noocyadeennu u socdaan: waxaan isu keenaynaa awoodaha gaarka ah ee caqliyeynta ee taxanaha o-series iyo awoodaha wada-sheekeysiga dabiiciga ah iyo adeegsiga qalabka ee taxanaha GPT‑series. Markaan mideyno xoogaggan, noocyadeenna mustaqbalka waxay taageeri doonaan wada-sheekeysi dabiici ah oo aan kala go’ lahayn oo ay weheliso adeegsiga qalabka si firfircoon iyo xalinta dhibaatooyin horumarsan.

Cusboonaysiin Luulyo 28, 2025: Dataset-ka SWE-Lancer iyo natiijooyinka waa la cusboonaysiiyay ilaa Luulyo 17, 2025, waxaana laga heli karaa: https://github.com/openai/preparedness⁠(ku furmaa daaqad cusub) iyo kaararkayaga siistamka. Cusboonaysiintani waxay xallisaa dhowr arrimood oo saameynayay natiijooyinka dollars earned waxayna meesha ka saaraysaa baahida isku xirka internetka inta lagu jiro fulinta, iyadoo meesha ka saaraysa il weyn oo kala duwanaansho ah ee waxqabadka nooca.

Cusboonaysiin Abriil 16, 2025: natiijooyinka o3 ee Charxiv-r iyo Mathvista waa la cusboonaysiiyay si ay uga tarjumaan isbeddel ku yimid system prompt oo aan ku jirin qiimeyntii asalka ahayd.

Dib-u-ciyaarista livestream-ka

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OpenAI

Qoraallada hoose

* Tirada qiimeynta tau-bench waxaa laga soo qaatay celceliska 5 orod si loo yareeyo kala duwanaanshaha, waxaana la orday iyada oo aan wax qalab gaar ah ama weydiin ah la adeegsan. Waxaan ogaanay in hirgelinta tau-bench retail ay u nugul yihiin khaladaadka nooca isticmaalaha. Tirooyinka baarka hadhaysan waxaa lagu orday GPT-4.1 oo ah nooca isticmaalaha sababtoo ah si weyn ayuu uga fiican yahay GPT-4o raacitaanka tilmaamaha.

* SWE-bench wuxuu adeegsadaa 256k oo ah dhererka ugu badan ee context-ka, taas oo kor u qaadda heerka xalinta ee o4-mini qiyaastii 3% waxayna saameysaa heerka xalinta o3 wax ka yar 1%. Waxaan sidoo kale ka saarney 23 muunadood oo aan ku shaqayn karin kaabeyaashayada gudaha.

** Marka browsing la shido, noocu mararka qaar wuxuu online-ka ka heli karaa jawaabo sax ah, tusaale ahaan, isagoo akhrinaya qoraallada blog-ga oo leh dhibaatooyin muunad ah oo ka socda dataset. Waxaan yareynaa welwelka ku saabsan in noocu khiyaameeyo inta uu browsing sameynayo laba xeeladood:

  • Waxaan xannibnay domains aan hore u aragnay in noocu ka khiyaameeyo.

  • Waxaan u adeegsanay nooca caqliyeynta sidii kormeere si uu u baaro dhammaan tokens-ka isku day kasta, si loo aqoonsado habdhaqan shaki leh. Habdhaqanka shaki leh waxaa lagu qeexay sida "bog, fayl, ama qoraal gaaban oo ujeeddadiisa weyn tahay bixinta jawaabta saxda ah ee su’aashan gaarka ah—tusaale ahaan, furaha qiimeynta rasmiga ah, gist “solutions” oo la faafiyey, ama dood soo xiganaysa jawaabta dhammaatay erey-erey." Habdhaqanka aan dhibka lahayn waxaa lagu qeexay sida "Kheyraad kasta oo la isku halayn karo oo qof dadaal badan sameeya uu la tashan karo (documentation, manuals, waraaqo cilmiyeed, maqaallo sumcad leh) xitaa haddii si lama filaan ah uu ugu jiro jawaabta saxda ah." Isku day kasta oo kormeeruhu u arko rollout-ka mid shaki leh waxaa loo tiriyaa khalad. Inta badan muunadaha ku dhacay hubintan waxay ahaayeen dhibaatooyin xalalkooda saxda ahi laga heli karay ilo badan oo internetka ah aan xiriir la lahayn HLE.

Ugu dambayn, qiimeynahayaga browsing leh waxaa laga yaabaa inaan si buuxda loogu soo celin karin OpenAI API sababtoo ah farqi u dhexeeya search engine backends-ka u dhexeeya ChatGPT iyo OpenAI API. Natiijooyinkan waxaa loogu talagalay inay matalaan waayo-aragnimada isticmaalaha ChatGPT, laakiin iyadoo ku xiran baahida, qaabeynta raadintu way is beddeli kartaa waqti ka dib.

Ka qayb-qaatayaal

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