<prompt> Fana ka tlhaloso ea lipalo bakeng sa theorem ea ho qetela ea fermat.nan  <response>: Instruction in English: Provide a mathematical explanation for fermats last theorem. . Response in English: Fermats Last Theorem, also known as Fermats Conjecture, is a statement in number theory, first proposed by Pierre de Fermat in 1637. The theorem states that no three positive integers a, b, and c can satisfy the equation an bn cn for any integer value of n greater than 2. In other words, the equation has no whole number solutions for n 2.The proof of this theorem eluded mathematicians for over 350 years, with many attempts being made to prove it. In 1995, Andrew Wiles, a British mathematician, finally published a proof of Fermats Last Theorem, which drew upon many different areas of mathematics, including algebraic geometry, number theory, and representation theory.Wiles proof uses a concept called elliptic curves, which are equations of the form y2 x3 ax b, and their associated modular forms. The proof relies on a conjecture known as the Taniyama-Shimura-Weil conjecture, which states that every elliptic curve defined over the rational numbers is also a modular form, in a sense defined by the conjecture. Wiles was able to prove a special case of this conjecture, now known as the Modularity Theorem, which was sufficient to prove the theorem.In showing that a certain type of elliptic curve could not exist, Wiles demonstrated that there could be no solutions to Fermats equation for n 2. Thus, through his work, Andrew Wiles was able to prove what had long been considered one of the greatest unsolved problems in mathematics - Fermats Last Theorem. . Response in Sesotho: Fermats Last Theorem, eo hape e tsejoang e le Fermats Conjecture, ke polelo ea khopolo ea palo, e hlahisitsoeng pele ke Pierre de Fermat ka 1637. Khopolo-taba e bolela hore ha ho lipalo tse tharo tse nepahetseng a, b, le c tse ka khotsofatsang equation an b n cn bakeng sa boleng bofe kapa bofe bo felletseng ba n kholo ho feta 2. Ka mantsoe a mang, equation ha e na tharollo ea linomoro kaofela bakeng sa n 2. Bopaki ba khopolo ena bo ile ba hloloa ho litsebi tsa lipalo ka lilemo tse fetang 350, ka boiteko bo bongata bo ntseng bo etsoa ho paka. eona. Ka 1995, Andrew Wiles, setsebi sa lipalo sa Lebrithani, qetellong o ile a hatisa bopaki ba Theorem ea ho Qetela ea Fermat, e ileng ea ama likarolo tse ngata tse fapaneng tsa lipalo, ho kenyeletsoa algebraic geometry, theory of number, le theory of representation. Bopaki ba Wiles bo sebelisa mohopolo o bitsoang elliptic curves, e leng li-equation tsa sebopeho sa y2 x3 ax b, le mefuta ea tsona ea modular. Bopaki bo itšetlehile ka khopolo e tsejoang e le khopolo ea Taniyama-Shimura-Weil, e bolelang hore lekhalo le leng le le leng la elliptic le hlalosoang holima lipalo tse utloahalang le eona ke sebopeho sa modular, ka kutloisiso e hlalosoang ke khopolo. Wiles o ile a khona ho paka nyeoe e khethehileng ea khopolo ena, eo hona joale e tsejoang e le Modularity Theorem, e neng e lekane ho paka khopolo-taba. Ha a bontša hore mofuta o itseng oa elliptic curve o ne o ke ke oa ba teng, Wiles o ile a bontša hore ho ne ho ke ke ha e-ba le tharollo ea equation ea Fermat bakeng sa n 2. Kahoo, ka mosebetsi oa hae, Andrew Wiles o ile a khona ho paka seo e leng khale se nkoa e le se seng sa tse kholo ka ho fetisisa. mathata a sa rarolloang thutong ea lipalo - Fermats Last Theorem. 