Related papers: First Case of Fermat's Last Theorem
We show that an elementary proof of Fermat's Last Theorem (FLT) exists. Our paper also extends the scope of FLT from integers to all rational numbers.
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,…
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 $2p+1$…
From some works of P. Furtw\"angler 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…
We formalise the proof of the first case of Fermat's Last Theorem for regular primes using the \emph{Lean} theorem prover and its mathematical library \emph{mathlib}. This is an important 19th century result that motivated the development…
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…
We propose a new approach at Fermat's Last Theorem (FLT) solution: for each FLT equation we associate a polynomial of the same degree. The study of the roots of the polynomial allows us to investigate the FLT validity. This technique,…
An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of…
This article, complement to the article [Que], deals with some generalizations of Futw\"angler's theorems for the second case of Fermat's Last Theorem (FLT2). Let $p$ be an odd prime, $\zeta$ a $p$th primitive root of unity, $K:=\Q(\zeta)$…
We show that Fermat's last theorem and a combinatorial theorem of Schur on monochromatic solutions of $a+b=c$ implies that there exist infinitely many primes. In particular, for small exponents such as $n=3$ or $4$ this gives a new proof of…
A elementary proof of Fermat"s Last Theorem[1] is presented for the case of even exponents n=2q, where q is any integer, including 2. For even exponents, the proof of the theorem reduces to showing that solutions of the Pythagorean equation…
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…
The main result of the present article is a proof of Fermat's Last Theorem for sufficiently large prime exponents $p$ with $p \equiv 2 \pmod{3}$ over certain number fields. A particular case of these fields are the maximal real subfields of…
Polignac [1] conjectured that for every even natural number $2k (k\geq1)$, there exist infinitely many consecutive primes $p_n$ and $p_{n+1}$ such that $p_{n+1}-p_n=2k$. A weakened form of this conjecture states that for every $k\geq1$,…
If $p_k$ is the k-th prime, the Firoozbakht conjecture states that the sequence $(p_k)^{1/k}$ is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify…
In the present paper we study, in a mathematically non-formal way, the validity of the Fermat's Last Theorem (FLT) by generalizing the usual procedure of extracting the square root of non convenient objects initially introduced by P. A. M.…
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 $a^p+b^p+c^p=0$…
A suggestion is put forward regarding a partial proof of FLT(case1), which is elegant and simple enough to have caused Fermat's enthusiastic remark in the margin of his Bachet edition of Diophantus' "Arithmetica". It is based on an…
We announce here that Fermat's Last theorem was solved, but there is an easy proof of it on the basis of elemetary undergraduate mathematics. We shall disclose such an easy proof.
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…