Binomial Coefficients, Roots of Unity and Powers of Prime Numbers

2017

We study the solutions of certain congruences in different rings. The congruences include

Pacific Journal of Mathematics, 1969

A classical theorem discovered independently by J. Sylvester and I. Schur states that in a set of k consecutive integers, each of which is greater than k 9 there is a number having a prime divisor greater than k. In giving an elementary proof, P. Erdόs expressed the theorem in the following form:

Acta Arithmetica, 2003

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).

Acta Arithmetica, 1994

In this paper, the author presents a special polynomial function ( ) 1 ( ) ( ) a P r n n r r fx x r φ = = − ∏ , { } 1 ( ) , , a P r r rφ ∈ L , φ is Euler's function. 1 ( ) , , a P r rφ L are integer numbers relatively prime to the a P and n is odd integer, then he obtains its value congruent modulo a P where P and a denote an odd optional prime and a natural number respectively so ( 1) ( 1) ( ) ( 1) (mod ) a P n P n a f x x P − − ⎡ ⎤ ≡ + − ⎣ ⎦ , x Z ∈ and ( ) [ ] f x Z x ∈ .Here, the extended coefficients of the function, summation, and multiplication of all the members of the reduced residue system congruent modulo a P are also obtained. A special problem is proved by two methods: 1-multiplicative method 2-using groups and rings theory.

2003

Bulletin of the Polish Academy of Sciences Mathematics, 2004

What should be assumed about the integral polynomials f 1 (x),. .. , f k (x) in order that the solvability of the congruence f 1 (x)f 2 (x) • • • f k (x) ≡ 0 (mod p) for sufficiently large primes p implies the solvability of the equation f 1 (x)f 2 (x) • • • f k (x) = 0 in integers x? We provide some explicit characterizations for the cases when f j (x) are binomials or have cyclic splitting fields.

HAL (Le Centre pour la Communication Scientifique Directe), 2012

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).

In this paper we collected problems, which was either proposed or follow directly from results in our papers.