Related papers: Fermat's Last Theorem for regular primes
This expository and review paper deals with the Diamond Lemma for ring theory, which is proved in the first section of G. M. Bergman, The Diamond Lemma for Ring Theory, Advances in Mathematics, 29 (1978), pp. 178-218. No originality of the…
Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an…
We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…
Formula for the nth prime using elementary arithmetical functions based in a previous formula changing the characteristic function of prime numbers.
In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions.…
We prove that neither a prime nor {an l-almost prime} number theorem hold in the class of regular Toeplitz subshifts. But, {when a quantitative strengthening of the regularity with respect to the periodic structure involving Euler's totient…
We give a proof of Fermat's little theorem which does not use nor arithmetic(Euclidean algorithm) neither algebra (group theory), but it rather employs the field of the formal power series Q((x)). The note is an example of a mathematical…
We show that the classical discrete logarithm problem over prime fields can be reduced to that of solving a system of linear modular equations.
In the last few years there have been rapid developments in SMT solving for finite fields. These include new decision procedures, new implementations of SMT theory solvers, and new software verifiers that rely on SMT solving for finite…
In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is…
A classical result in number theory is Dirichlet's theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k>1. Building upon a proof by E.M. Wright in 1954, we…
This paper explores some previously-unrecognized consequences of Lerch's 1905 formula for the Fermat quotient, with special attention to the sums which he introduced in this context. A generalization of his result is proved, and a new proof…
A key ingredient in the Taylor-Wiles proof of Fermat last theorem is the classical Ihara's lemma which is used to rise the modularity property between some congruent galoisian representations. In their work on Sato-Tate,…
In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm…
We prove explicit versions of Cram\'er's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.
It is still open whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning the Mersenne numbers is also unsolved. Extending some results from [9], we characterizethe the…
We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any $M>0$, any finite field…
The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Fr\"uhwirth, we consider an extension of this theory with an additional…
Two well known facts from elementary number theory are proven by using Bergman spaces.
In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to…