Nuno Freitas, Alain Kraus, Samir Siksek
Let F be a totally real number field of odd degree. We prove several purely local criteria for the asymptotic Fermat's Last Theorem to hold over F, and also for the non-existence of solutions to the unit equation over F. For example, if 2 totally ramifies and 3 splits completely in F, then the asymptotic Fermat's Last Theorem holds over F.
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In our work we give the examples using Fermat's Last Theorem for solving some problems from algebra and number theory.
This is a survey article on developments in modularity since the proof of Fermat's Last Theorem, with an emphasis on the historical development of the subject rather than any technical details.
This paper has been withdrawn by the author [arXiv admin].
Assuming two deep but standard conjectures from the Langlands Programme, we prove that the asymptotic Fermat's Last Theorem holds for imaginary quadratic fields Q(\sqrt{-d}) with -d=2, 3 mod 4. For a general number field K, again assuming standard conjectures, we give a criterion based on the solutions to a certain S-unit equation, which if satisfied implies the asymptotic Fermat's Last Theorem.
Recent work of Freitas and Siksek showed that an asymptotic version of Fermat's Last Theorem holds for many totally real fields. Later this result was extended by Deconinck to generalized Fermat equations of the form $Ax^p +By^p +Cz^p = 0$, where A;B;C are odd integers belonging to a totally real field. Another extension was given by Sengun and Siksek. They showed that the Fermat equation holds asymptotically for imaginary quadratic number fields satisfying usual conjectures about modularity. In this work, combining their techniques we extend their results about the generalized Fermat equatio…
Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from class field theory, the theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number fields, and for thousands of number fields of small degree. For example, we prove the effect…
Let $F$ be a number field and $\mathcal{O}_F$ its ring of integers. We use Chevalley's ambiguous class number formula to give a criterion for the non-existence of solutions to the unit equation $λ+ μ= 1$, $λ, μ\in \mathcal{O}_F^\times$. This is then used to strengthen a criterion for the asymptotic Fermat's Last Theorem due to Freitas and Siksek.
Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n=3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.
Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no non-trivial solutions over the quadratic field $\mathbb{Q}(\sqrt{d})$ for $n \ge 4$. Furthermore, we show for $d=17$ that the same holds for prime exponents $n \equiv 3$, $5 \pmod{8}$.
`Fermat's Last Theorem for the exponent 3 has received numerous proofs, the most common of which being either in Euler's or in Gauss' style. This latter works entirely in the ring of integers of the quadratic field generated by the square root of -3. A proof in Euler's style is based on a central lemma related to properties of the quadratic form x^2 + 3y^2, then it proceeds by descent by considering Two main cases. In the present version, the central lemma receives a short proof and the classical proof by descent in Two main cases boils down to a One-case proof.
Fermat's Last Theorem over $\mathbb{Q}(\sqrt{2},\sqrt{3})$
研究实双二次域上的费马方程，证明在 $\mathbb{Q}(\sqrt{2},\sqrt{3})$ 上 $n\geq 4$ 时无非平凡解。
Let $K$ be a totally real field. By the asymptotic Fermat's Last Theorem over $K$ we mean the statement that there is a constant $B_K$ such that for prime exponents $p>B_K$ the only solutions to the Fermat equation $a^p + b^p + c^p = 0$ with $a$, $b$, $c$ in $K$ are the trivial ones satisfying $abc = 0$. With the help of modularity, level lowering and image of inertia comparisons we give an algorithmically testable criterion which if satisfied by $K$ implies the asymptotic Fermat's Last Theorem over $K$. Using techniques from analytic number theory, we show that our criterion is satisfied by $…
In this paper we study the Fermat equation $x^n+y^n=z^n$ over quadratic fields $\mathbb{Q}(\sqrt{d})$ for squarefree $d$ with $26 \leq d \leq 97$. By studying quadratic points on the modular curves $X_0(N)$, $d$-regular primes, and working with Hecke operators on spaces of Hilbert newforms, we extend work of Freitas and Siksek to show that for most squarefree $d$ in this range there are no non-trivial solutions to this equation for $n \geq 4$.
We prove Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$ and ${\mathbb Q}(\sqrt{17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer-Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of ${\mathbb Q}(\sqrt{5})$ is a generalization to a real quadratic base field of the one used by Chen-Siksek. For the case of ${\mathbb Q}(\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbe…
On Fermat's Last Theorem over the $\mathbb{Z}_3$-extension of $\mathbb{Q}$ and other fields
本文结合模方法与 Pomey 算术结果的推广，证明了在包括分圆域极大实子域在内的某些数域上，满足 $p\equiv2\pmod3$ 的充分大素指数情形的费马大定理。
The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$ is a prime number, the first case of Fermat's Last Theorem is true for $p$. In this paper, we investigate such obstructions over number fields. We obtain analogous results on Sophie Germain type criteria, for imaginary quadratic fields. Furthermore, extending a well known statement over ${\bf Q}$, we give an easily testable condition which allows occasionally…
We give again the proof of several classical results concerning the cyclotomic approach to Fermat's last theorem using exclusively class field theory (essentially the reflection theorems), without any calculations. The fact that this is possible suggests a part of the logical inefficiency of the historical investigations. We analyze the significance of the numerous computations of the literature, to show how they are probably too local to get any proof of the theorem. However we use the derivation method of Eichler as a prerequisite for our purpose, a method which is also local but more effect…
This paper has been withdrawn by the author for further investigation.
Let $p$ be an odd prime number. Using modular arguments, we give an easy testable condition which allows often to prove Fermat's Last Theorem over the quadratic field ${\bf Q}(\sqrt{5})$ for the exponent $p$. It is related to the Wendt's resultant of the polynomials $X^n-1$ and $(X+1)^n-1$. We deduce Fermat's Last Theorem over this field in case one has $5\leq p<10^7$, and we obtain analogous results on Sophie Germain type criteria.
In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have in fact been proved not to have any unknown solutions, these improved bounds are interesting in the context of a new effective abc inequality announced in the paper \cite{MFHMP} based on Mochizuki's \cite{Mo}[IUT-IV, Theorem A]. In this paper we provide a strengthening of the lower bound for FLT2, which is necessary in order to take advantage of the best up…
In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begin{equation} \begin{cases} A+B=C ABC=D^n \end{cases} \end{equation} where $A,B,C,D,n \in\ZZ_{+}$ and $n\geq3$, which may be regarded as a generalization of Fermat's equation $x^n+y^n=z^n$. When $\gcd(A,B,C)=1$, $(1)$ is equivalent to Fermat's equation, which means it has no positive integer solutions. We discuss several cases for $\gcd(A,B,C)=p^k$ where $p$ is an odd prime. In particular, for $k=1$ we prove that …
Fermat's Last Theorem for Selmer sections
本文证明了对于指数ℓ≥7的仿射费马曲线，所有定义在有理数上的塞尔默截面都是尖点的。
We study Kummer's approach towards proving the Fermat's last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other things the Class number formula, and use this formula to conclude our study.
We formulate an exponential Diophantine equation, which is is some sense one order higher that Fermat's Last Theorem. We also give three examples of solutions to this exponential Diophantine equation and formulate a conjecture.
Asymptotic Fermat's Last Theorem for a family of equations of signature $(2, 2n, n)$
针对签名为$(2, 2n, n)$的费马型方程$Cx^2 + q^ky^{2n} = z^n$，研究人员借助模方法、水平降低与伽罗瓦理论，给出可算法检验的条件，证明满足条件时存在常数$B_{C,q}$，当$n$大于该常数时方程无整数解。
This paper is submitted to Algebraic-Number-Theory Archives for validation by Number Theorists Community. It is an update of the previous versions ANT-0155, ANT-0170, ANT-0205, ANT-0237, ANT-0321, ANT-0333, and ANT-0356, of which the first four were titled `A generalization of Eichler criterium for Fermat's Last Theorem' and the last three were titled `A classical approach on Fermat-Wiles theorem'. This version contains a complete reorganization of the paper with a first part dealing with cyclotomic fields (independently of FLT) from page 10 to 72 and a second part dealing with FLT from page 7…
We present the proof of Diophantus' 20th problem (book VI of Diophantus' Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used the "wonderful" method (ipse dixit Fermat) of infinite descent. This method, which is, historically, the first use of induction, consists in producing smaller and smaller non-negative integer solutions assuming that one exists; this naturally leads to a reductio ad absurdum reasoning because we are boun…
Let $K$ be a totally real number field. For all prime number $p\geq 5$, let us denote by $F_p$ the Fermat curve of equation $x^p+y^p+z^p=0$. Under the assumption that $2$ is totally ramified in $K$, we establish some results about the set $F_p(K)$ of points of $F_p$ rational over $K$. We obtain a criterion so that the asymptotic Fermat's Last Theorem is true over $K$, criterion related to the set of Hilbert modular cusp newforms over $K$, of parallel weight $2$ and of level the prime ideal above $2$. It is often simply testable numerically, particularly if the narrow class number of $K$ is $1$…
In a recent paper, Freitas and Siksek proved an asypmtotic version of Fermat's Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A x^p+B y^p+ C z^p=0$, where $A$, $B$, $C$ are odd integers belonging to a totally real field.
Fermat-Catalan and Tijdeman-Zagier conjectures for products
本文将 Fermat-Catalan 猜想、Tijdeman-Zagier 猜想与 Fermat 大定理推广到整数乘积情形，并给出 abc 猜想的新显式表述。
A complete formalization of Fermat's Last Theorem for regular primes in Lean
本研究在Lean4定理证明器中完成费马大定理正则素数情形的完整形式化证明，借助Hilbert第90-94定理证明核心的Kummer引理，相较类域论的现代证法更适配形式化需求。
From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) = A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then Fermat's last theorem holds for p. More generally, t…
Let $K$ be a number field and $p$ a prime number $\geq 5$. Let us denote by $μ_p$ the group of the $p$th roots of unity. We define $p$ to be $K$-regular if $p$ does not divide the class number of the field $K(μ_p)$. Under the assumption that $p$ is $K$-regular and inert in $K$, we establish the second case of Fermat's Last Theorem over $K$ for the exponent $p$. We use in the proof classical arguments, as well as Faltings' theorem stating that a curve of genus at least two over $K$ has a finite number of $K$-rational points. Moreover, if $K$ is an imaginary quadratic field, other than ${\bf Q}(…
Fermat's Last Theorem Implies Euclid's Infinitude of Primes
该研究借助费马大定理与舒尔组合定理给出小指数情形下欧几里得素数无穷性定理的新证明，还探讨了多个组合数论结果对该定理的蕴含，揭示其为诸多数学结果的必要条件。
Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $μ$ is a zero of $X_n$ if and only if the triple $(μ,μ+m,μ+\ell)$ satisfies $x^n + y^n = z^n$. It is shown that the irreducibility of these polynomials implies Fermat's last theorem. It is also shown, in a precise asymptotic sense, that for a vast majority of cases, these polynomials are irreducible via Eisenstein's criterion. We conclude by offering a conjecture on powerful numbers.
Beyond and slightly aside of the Fermat Desert
本文利用分圆方法证明一类广义费马方程在互素整数及素数 p>3 条件下无解，并由此推出费马大定理。
Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat's Last Theorem over $\mathbb Q(i)$. Under the same assumption, we also prove that, for all prime exponents $p \geq 5$, Fermat's equation $a^p+b^p+c^p=0$ does not have non-trivial solutions over $\mathbb Q(\sqrt{-2})$ and $\mathbb Q(\sqrt{-7})$.
Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem
研究有理数a下多项式x^n+(1-x)^n+a^n的不可约性，证明多个无穷族的不可约性，并分析复根位置，同时揭示a=±1时与Cauchy-Mirimanoff多项式及Nanninga多项式的因式分解关系。
Fermat's Last Theorem for regular primes
本文使用Lean定理证明器及其数学库mathlib，形式化了正则素数的费马大定理第一情形的证明，并分析了形式化过程中的困难及解决方案。
A survey on the generalized Fermat equation of various signatures over totally real fields
本文综述了在完全实数域上使用模性方法研究不同签名的推广费马方程的最新研究成果。
This article deals with a conjecture, introduced in [GQ] (hereinafter $SFLT2$), which generalizes the second case of Fermat's Last Theorem: {\it Let $p>3$ be a prime. The diophantine equation $\frac{u^p+v^p}{u+v}=w_1^p$ with $u,v,u+v, w_1\in\Z\backslash\{0\}$, $u,v$ coprime and $v\equiv 0 \mod p$ has no solution.} Let $ζ$ be a $p$th primitive root of unity and $K:=\Q(ζ)$. A prime $q$ is said {\it $p$-principal} if the class of any prime ideal $\mathfrak q_K$ of $K$ over $q$ is a $p$-power of a class. Assume that $SFLT2$ fails for $(p,u,v)$. Let $q$ be any odd prime coprime with $puv$, $f$ the …
We solve the diophantine equations x^4 + d y^2 = z^p for d=2 and d=3 and any prime p>349 and p>131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermat's Last Theorem, and by Ellenberg in the solution of the equation x^4 + y^2 = z^p, and we use Q-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations for other values of d.
In a recent work the authors prove the effective asymptotic Fermat's Last Theorem for the infinite family of fields $\mathbb{Q}(ζ_{2^{r+2}})^+$ where $r \ge 0$. A crucial step in their proof is the following conjecture of Kraus. Let $K$ be a number field having odd narrow class number and a unique prime $λ$ above $2$. Then there are no elliptic curves defined over $K$ with conductor $λ$ and a $K$-rational point of order $2$. In this note we give a new elementary proof of Kraus' conjecture that makes use only of basic facts about elliptic curves, Tate curves and Tate modules.
Let $p \ge 5$ be a prime and let $\mathbb{Q}_{n,p}$ denote the $n$-th layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. We show that $\mathbb{Q}_{n,p}$ has no exceptional units. We use this to prove the effective asymptotic Fermat's Last Theorem over $\mathbb{Q}_{n,p}$ for all $n \ge 1$ and all primes $p \ge 5$ that are non-Wieferich, i.e. $2^{p-1} \not \equiv 1 \pmod{p^2}$. The effectivity in our result builds on recent work of Thorne proving modularity of elliptic curves over $\mathbb{Q}_{n,p}$.
This article, complement to the article [Que], deals with some generalizations of Futwängler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $ζ$ a $p$th primitive root of unity, $K:=\Q(ζ)$ and $C\ell_K$ the class group of $K$. A prime $q$ is said $p$-principal if the class $c\ell_K (\mk q_K)\in C\ell_K$ of any prime ideal $\mk q_K$ of $\Z_K$ over $q$ is the $p$th power of a class. Assume that FLT2 fails for $(p,x,y,z)$ where $x, y, z$ are mutually coprime integers, $p$ divides $y$ and $x^p+y^p+z^p=0$. Let $q$ be a prime dividing $\frac{(x^p+y^p)(y^p+z^p…
Sophie Germain, mathématicienne extraordinaire: A story stranger than fiction
本文回顾索菲·热尔曼在数论领域的贡献，揭示她对费马大定理的深入研究及与拉格朗日、勒让德和高斯的互动，重新评估其在数论中的重要性。
We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved in coprime integers, where $p \ge 7$ is prime. Using modularity and level lowering techniques, the problem can be reduced to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We first develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minima…
Comments about the paper by Elsholz, Fermat's last theorem implies Euclid's infinitude of primes, (2021), and simplification.
Asymptotic solutions of the generalized Fermat-type equation of signature $(p,p,3)$ over totally real number fields
本文研究了全实数域上签名 $(p,p,3)$ 的广义费马型方程 $Ax^p+By^p=Cz^3$ 的渐近解，证明了在特定条件下该方程无解，并给出了若干局部判据。
Fermat near misses and the integral Hilbert Property
本文研究了四次费马方程的近似解问题，通过德尔皮佐表面几何学证明了积分点非稀疏性。
On Darmon's program for the generalized Fermat equation, I
本文基于过去二十年模方法进展，扩展Darmon研究广义费马方程的纲领，开发Frey阿贝尔簇相关工具，处理特定签名方程模方法步骤，解决部分方程解问题并推进纲领未处理的困难。
Let K be a totally real Galois number field and let A be a set of elliptic curves over K. We give sufficient conditions for the existence of a finite computable set of rational primes P such that for p not in P and E in A, the representation on E[p] is irreducible. Our sufficient conditions are often satisfied for Frey elliptic curves associated to solutions of Diophantine equations; in that context, the irreducibility of the mod p representation is a hypothesis needed for applying level-lowering theorems. We illustrate our approach by improving on an existing result for Fermat-type equations …
On the solutions of the generalized Fermat equation over totally real number fields
本文研究了总实数域上广义费马方程的渐近解问题，证明了在特定条件下该方程无渐近解，并计算了实二次域中满足条件的数域密度。
$7$-adic Galois representations of elliptic curves over the rationals via Kummer descent
本文通过 Kummer 下降证明了两个亏格 9 的模曲线无非 CM 有理点，从而完成了非 CM 椭圆曲线 $7$-adic Galois 像的分类，其关键依赖于指数为 7 的费马大定理。
A multi-Frey approach to Fermat equations of signature $(r,r,p)$
本文通过全实域上 Frey 椭圆曲线的模方法与精细化的多重 Frey 技术，解决了签名 $(r,r,p)$ 的两类广义费马方程，并给出 $x^5+y^5=d z^n$ 在局部条件下的新结果。
On Modular Approach to Diophantine Equation $x^4-y^4=nz^p$ over Number Fields
本文利用模方法证明了丢番图方程 $x^4-y^4=nz^p$ 在各类数域上解的渐近结果，并证明了方程 $x^4-y^4=2^αz^p$ 的渐近广义费马定理在无限多个二次数域上成立。
We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >> n^2. In other words, we prove a weak form of the Erdos-Szemeredi sum-product conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials. We also prove an analogue of a theorem of Bourgain and Chang for the ring C[x].
Generalized Fermat equation over cyclotomic $\mathbb{Z}_l$-extensions of totally real fields
研究全实域分圆 $\mathbb{Z}_l$-扩张各层上的渐近费马大定理和广义费马方程无解的条件，利用非Wieferich素数和椭圆曲线模性等工具。
The modular approach to Diophantine equations over totally real fields
本文综述了将Wiles证明Fermat大定理的模方法推广至全实数域上丢番图方程时所面临的新障碍，并调查了该领域现有的方法与结果。
Monodromy groups and exceptional Hodge classes, I: Fermat Jacobians
本文计算了超椭圆曲线y^2=x^m+1的Jacobian簇J_m的等元分解、端元定义域和连通单值域，并展示了高维退化现象。
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