Polynomials generalizing binomial coefficients and their application to the study of Fermat's last theorem (original) (raw)

Related papers

On the cyclotomic identity and related product expansions

Australasian Journal of …, 1993

The cy<:101GOIJ01C identity, that 1 1-az 00 n=l ( _1_) M(o:,n) 1 zn ' where M(a,n) = *-:L:dlnJ.L (~) ad, and J. L is the classical Mobius function, has several natural analogues. Polynomials in a of degree n related to M( a, n) in these identities share interesting properties with M( a, n).

Analysis of the classical cyclotomic approach to fermat's last theorem

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 effective. Then we propose some modest ways of study in a more diophantine context using radicals; this point of view would require further nonalgebraic investigations.

Vandiver papers on cyclotomy revisited and Fermat Last Theorem (avec Roland Quême)

Relying on classical studies of H.S. Vandiver and P. Furtwängler, we intend to lay the foundations of a new global cyclotomic approach to Fermat's Last Theorem (FLT) for p > 3 and to a stronger version called " Strong Fermat's Last Theorem " (SFLT), by introducing an infinite number of auxiliary cyclotomic fields of the form Q(µq−1) for q = p a prime.

Binomial Coefficients, Roots of Unity and Powers of Prime Numbers

Bulletin of the Malaysian Mathematical Sciences Society, 2022

Let t ∈ N + be given. In this article we are interested in characterizing those d ∈ N + such that the congruence 1 t t−1 s=0 n + dζ s t d − 1 ≡ n d − 1 (mod d) is true for each n ∈ Z. In particular, assuming that d has a prime divisor greater than t, we show that the above congruence holds for each n ∈ Z if and only if d = p r , where p is a prime number greater than t and r ∈ {1,. .. , t}.

On the arithmetic of generalized Fekete polynomials

arXiv (Cornell University), 2022

For each prime number p one can associate a Fekete polynomial with coefficients −1 or 1 except the constant term, which is 0. These are classical polynomials that have been studied extensively in the framework of analytic number theory. In a recent paper, we showed that these polynomials also encode interesting arithmetic information. In this paper, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be composite numbers. We then investigate the appearance of cyclotomic factors in these generalized Fekete polynomials. Based on this investigation, we introduce a compact version of Fekete polynomials as well as their trace polynomials. We then study the Galois groups of these Fekete polynomials using modular techniques. In particular, we discover some surprising extra symmetries which imply some restrictions on the corresponding Galois groups. Finally, based on both theoretical and numerical data, we propose a precise conjecture on the structure of these Galois groups.

Vandiver papers on cyclotomy revised and Fermat’s Last Theorem

Publications Mathématiques de Besançon

Relying on classical studies of H.S. Vandiver and P. Furtwängler, we intend to lay the foundations of a new global cyclotomic approach to Fermat's Last Theorem (FLT) for p > 3 and to a stronger version called " Strong Fermat's Last Theorem " (SFLT), by introducing an infinite number of auxiliary cyclotomic fields of the form Q(µq−1) for q = p a prime. We show that the existence of nontrivial counterexamples to SFLT implies strong constraints on the arithmetic of the fields Q(µq−1) with respect to Čebotarev's density theorem in suitable canonical Abelian p-extensions. Further investigations (of an analytic or a geometric nature) would be necessary to lead to a proof of SFLT. Our results imply sufficient conditions for the non-existence of nontrivial solutions of the SFLT equation and suggest various conjectures. We prove for instance that if there exist infinitely many primes q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2) such that for q | q in Q(µq−1), q 1−c is of the form a p (α) for some ideal a and some α ≡ 1 (mod p 2) (where c is the complex conjugation), then Fermat's Last Theorem holds for p. Résumé.-À partir de travaux classiques de H.S. Vandiver et P. Furtwängler, nous posons les bases d'une nouvelle approche cyclotomique globale du dernier théorème de Fermat pour p > 3 et d'une version plus forte appelée " Strong Fermat's Last Theorem " (SFLT), en introduisant une infinité de corps cyclotomiques auxiliaires de la forme Q(µq−1) pour q = p premier. Nous montrons que l'existence de contre-exemples non triviaux à SFLT implique de fortes contraintes sur l'arithmétique des corps Q(µq−1) au niveau du théorème de densité de Čebotarev dans certaines p-extensions abéliennes canoniques. Des investigations supplémentaires (analytiques ou géométriques) seraient nécessaires pour conduire à une preuve de SFLT. À partir de là, nous donnons des conditions suffisantes de non existence de solutions non triviales à l'équation associée à SFLT et formulons diverses conjectures. Nous prouvons par exemple que s'il existe une infinité de nombres premiers q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2), tels que pour q | q dans Q(µq−1), on ait q 1−c = a p (α) avec α ≡ 1 (mod p 2) (où c est la conjugaison complexe), alors le dernier théorème de Fermat est vrai pour p.

Some works of Furtw\"angler and Vandiver revisited and the Fermat last theorem

arXiv (Cornell University), 2011

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(µ q−1) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2), such that for q | q in Q(µ q−1), we have q 1−c = a p (α) with α ≡ 1 (mod p 2) (where c is the complex conjugation), then Fermat ′ s last theorem holds for p. More generally, the main purpose of the paper is to show that the existence of nontrivial solutions for SFLT implies some strong constraints on the arithmetic of the fields Q(µ q−1). From there, we give sufficient conditions of nonexistence that would require further investigations to lead to a proof of SFLT, and we formulate various conjectures. This text must be considered as a basic tool for future researches (probably of analytic or geometric nature). Résumé. Reprenant des travaux de P. Furtwängler et H.S. Vandiver, nous posons les bases d'une nouvelle approche cyclotomique du dernier théorème de Fermat pour p > 3 et d'une version plus forte appelée SFLT, en introduisant des corps gouvernants de la forme Q(µ q−1) pour q premier. Nous prouvons par exemple que s'il existe une infinité de nombres premiers q, q ≡ 1 (mod p), q p−1 ≡ 1 (mod p 2), tels que pour q | q dans Q(µ q−1), on ait q 1−c = a p (α) avec α ≡ 1 (mod p 2) (où c est la conjugaison complexe), alors le théorème de Fermat est vrai pour p. Plus généralement, le but principal de l'article est de montrer que l'existence de solutions non triviales pour SFLT implique de fortes contraintes sur l'arithmétique des corps Q(µ q−1). A partir de là, nous donnons des conditions suffisantes de non existence qui nécessiteraient des investigations supplémentaires pour conduireà une preuve de SFLT, et nous formulons diverses conjectures. Ce texte doitêtre considéré comme un outil de base pour de futures recherches (probablement analytiques ou géométriques). ******* This second version includes some corrections in the English language, an in depth study of the case p = 3 (especially Theorem 8), further details on some conjectures, and some minor mathematical improvements. 1 Equation (u + v ζ) Z[ζ] = w p 1 or p w p 1 , in integers u, v with g.c.d. (u, v) = 1, equivalent to N K/Q (u + v ζ) = w p 1 or p w p 1 , where ζ := e 2iπ/p , K := Q(ζ), p := (ζ − 1) Z[ζ] (see Conjecture 1). Remark that the important condition g.c.d. (u, v) = 1 implies w 1 prime to p. Note that if u v = 0, the condition g.c.d. (u, v) = 1 implies (u, v) = (±1, 0) or (0, ±1). 2 If ν ≥ 1, then α := a+c ζ a+c ζ −1 is a pseudo-unit (i.e., the pth power of an ideal), congruent to 1 modulo p; so, from [Gr1, Theorem 2.2, Remark 2.3, (ii)], α is locally a pth power in K giving easily α ≡ 1 (mod p p+1), then c (ζ−ζ −1) a+c ζ −1 ≡ 0 (mod p p+1), hence c ≡ 0 (mod p 2). 3 If u − v ≡ 0 (mod p), then α := uζ+v u+vζ is a pseudo-unit congruent to 1 modulo p; so, from [Gr1, Theorem 2.2, Remark 2.3, (ii)], α is locally a pth power giving α ≡ 1 (mod p p+1), then (u−v)(ζ−1) u+vζ ≡ 0 (mod p p+1), hence u − v ≡ 0 (mod p 2). This is valid in the Fermat case if x − z ≡ 0 (mod p), and gives x − z ≡ 0 (mod p 2).

N ov 2 01 9 Coefficients of ( inverse ) unitary cyclotomic polynomials

2019

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials Φ n pxq. They can be written as certain products of cyclotomic poynomials. We study the case where n has two or three distinct prime factors using numerical semigroups, respectively Bachman’s inclusion-exclusion polynomials. Given m ě 1 we show that every integer occurs as a coefficient of Φ mn pxq for some n ě 1 following Ji, Li and Moree [9]. Here n will typically have many different prime factors. We also consider similar questions for the polynomials pxn ́ 1q{Φ n pxq, the inverse unitary cyclotomic polynomials.