Showing papers on "Prime number published in 2006"
Book•
30 Jan 2006TL;DR: Sieve theory has a rich and romantic history as discussed by the authors, and Goldbach's conjecture that every even number can be written as the sum of two prime numbers has been two of the problems that have inspired the development of the theory.
Abstract: Sieve theory has a rich and romantic history. The ancient question of whether there exist infinitely many twin primes (primes p such that p+2 is also prime), and Goldbach's conjecture that every even number can be written as the sum of two prime numbers, have been two of the problems that have inspired the development of the theory. This book provides a motivated introduction to sieve theory. Rather than focus on technical details which can obscure the beauty of the theory, the authors focus on examples and applications, developing the theory in parallel. The text can be used for a senior level undergraduate course or an introductory graduate course in analytic number theory, and non-experts can gain a quick introduction to the techniques of the subject.
113 citations
1 Jan 2006
TL;DR: Granville and Rudnick as discussed by the authors proposed a uniform distribution based on the Hardy-Littlewood circle method for quadratic numbers, and showed that the distribution can be computed in polynomial time.
Abstract: Preface. Contributors.- Biographical Sketches of the Lecturers. Uniform Distribution A. Granville and Z. Rudnick.- 1. Uniform Distribution mod One.2. Fractional Parts of an2 .3. Uniform Distribution mod N.4. Normal Numbers. Sieving and the Erdos-Kac Theorem A. Granville and K. Soundararajan.- Uniform Distribution, Exponential Sums, and Cryptography J. B. Friedlander .- 1 Randomness and Pseudorandomness. 2 Uniform Distribution and Exponential Sums. 3. Exponential Sums and Cryptography. 4. Some Exponential Sum Bounds. 5. General Modulus and Discrepancy of Diffie-Hellman Triples. 6. Pseudorandom Number Generation. 7 Large Periods and the Carmichael Function. 8 Exponential Sums to General Modulus. 9. Sums over Elliptic Curves. 10 Proof Sketch of Theorem 4.1.- The Distribution of Prime Numbers K. Soundararajan.- 1. The Cramer Model and Gaps Between Consecutive Primes. 2 The Distribution of Primes in Longer Intervals. 3 Maier's Method and an "Uncertainty Principle" .- Torsion Points on Curves A. Granville and Z. Rudnick.- 1. Introduction. 2. A Proof Using Galois Theory. 3. Polynomials Vanishing at Roots of Unity. The distribution of roots of a polynomial A. Granville.- 1. Introduction. 2 Algebraic Numbers. 3 In k Dimensions: the Bilu Equidistribution Theorem. 4. Lower Bounds on Heights. 5. Compact Sets with Minimal Energy.- Manin-Mumford, Andre-Oort, the Equidistribution Point of View Emmanuel Ullmo.- 1 Introduction.2 Informal Examples of Equi-Distribution.3. The Manin-Mumford and the Andre-Oort Conjecture. 4. Equidistribution of Special Subvarieties Analytic Methods for the Distribution of Rational Points on Algebraic Varieties D. R. Heath-Brown.- 1. Introduction to the Hardy-Littlewood Circle Method. 2. Major Arcs and Local Factors in the Hardy-Littlewood Circle Method. 3. The Minor Arcs in the Hardy-Littlewood Circle Method.4. Combining Analytic and Geometric Methods. Universal Torsors over Del Pezzo Surfaces and Rational Points U. Derenthal and Y.Tschinkel.- 1. Introduction. 2. Geometric Background. 3. Manin's Conjecture. 4. The Universal Torsor. 5. Summations.6. Completion of the Proof. 7. Equations of Universal Torsors.- An Introduction to the Linnik Problems W. Duke.- 1. Introduction.2. The Linnik Problems. 3. Holomorphic Modular Forms of Half-Integral Weight. 4. Theta Series With Harmonic Polynomials. 5. Linnik Problem for Squares and the Shimura Lift. 6. Nontrivial Estimates for Fourier Coefficients.7. Salie Sums. 8. An Estimate of Iwaniec. 9. Theorems of Gauss and Siegel . 10. The Nonholomorphic Case (Duke, 1988). 11. Transition to Subconvexity Bounds for L-Functions. 12. An Application to Traces of Singular Moduli. Distribution Modulo One and Ratner's Theorem J. Marklof.- 1. Introduction. 2. Randomness of Point Sequences mod 1. 3. ma mod One 4. vma mod One.5. Ratner's Theorem. Spectral Theory of Automorphic Forms: A Very Brief Introduction A. Venkatesh.- 1. What Is a Homogeneous Space?. 2. Spectral Theory: Compact Case. 3. Dynamics. 4. Spectral Theory: Noncompact Case. 5. Hecke Operators. 6. Gross Omissions: The Selberg Trace Formula. Some Examples How to Use Measure Classification in Number Theory E. Lindenstrauss.- 1. Introduction. 2. Dynamical Systems: Some Background. 3. Equidistribution of n2a mod 1. 4. Unipotent Flows and Ratner's Theorems. 5. Entropy of Dynamical Systems: Some More Background. 6. Diagonalizable Actions and the Set of Exceptions to Littlewood's Conjecture. 7. Applications to Quantum Unique Ergodicity.-An Introduction to Quantum Equidistribution S.De Bievre.- 1. Introduction. 2. A Crash Course in Classical Mechanics. 3.A Crash Course in Quantum Mechanics. 4. Two Words on Semi-Classical Analysis. 5. Quantum Mechanics on the Torus. The Arithmetic Theory of Quantum Maps Z. Rudnick.- 1. Quantum Mechanics on the Torus. 2. Quantizing Cat Maps. 3. Quantum Ergodicity. 4. Quantum Unique Ergodicity. 5. Arithmetic QUE .
99 citations
TL;DR: Goldston, Pintz, and Yildirim as discussed by the authors showed that there are infinitely many prime numbers for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes.
Abstract: In early 2005, Dan Goldston, Janos Pintz, and Cem Yildirim [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes. Before their work, it was known only that there were infinitely many gaps which were about a quarter the size of the average gap. The new result may be viewed as a step towards the famous twin prime conjecture that there are infinitely many prime pairs p and p+2, the gap here being 2, the smallest possible gap between primes. Perhaps most excitingly, their work reveals a connection between the distribution of primes in arithmetic progressions and small gaps between primes. Assuming certain (admittedly difficult) conjectures on the distribution of primes in arithmetic progressions, they are able to prove the existence of infinitely many prime pairs that differ by at most 16. The aim of this article is to explain some of the ideas involved in their work. Let us begin by explaining the main question in a little more detail. The number of primes up to x, denoted by π(x), is roughly x/ log x for large values of x; this is the celebrated Prime Number Theorem. Therefore, if we randomly choose an integer near x, then it has about a 1-in-log x chance of being prime. In other words, as we look at primes around size x, the average gap between consecutive primes is about log x. As x increases, the primes get sparser and the gap between consecutive primes tends to increase. Here are some natural questions about these gaps between prime numbers. Do the gaps always remain roughly about size log x, or do we sometimes get unexpectedly large gaps and sometimes surprisingly small gaps? Can we say something about the statistical distribution of these gaps? That is, can we quantify how often the gap is between, say, α log x and β log x, given 0 ≤ α < β? Except for the primes 2 and 3, clearly the gap between consecutive primes must be even. Does every even number occur infinitely often as a gap between consecutive primes? For example, the twin prime conjecture says that the
94 citations
TL;DR: In this article, Bruinier et al. gave a generalization of Gross and Zagier's work on singular moduli for the ring of integers and the conjugation in F.
Abstract: This is a joint work with Jan Bruinier, and is a generalization of the well-known work of Gross and Zagier on singular moduli [GZ]. Here is the main result. For detail, please see [BY]. Let p ≡ 1 (mod 4) be a prime number and F = Q(√p). We write OF for the ring of integers of F , and x 7→ x′ for the conjugation in F . Let Γ = SL2(OF ) be the Hilbert modular group associated to F . The corresponding Hilbert modular surface X = Γ\H2 is a normal quasi-projective algebraic variety defined over Q. Let K = F ( √ ∆) be a non-biquadratic quartic CM number field (containing F ) with discriminant dK = p q for some prime q ≡ 1 mod 4 (technical condition). Let σ and σ′ be the complex embeddings of K given by σ( √ ∆) = σ( √ ∆′), and σ( √ ∆) = −√∆′. Then Φ = {1, σ} and Φ′ = {1, σ′} are two CM types. Let CM(K, Φ) be the (formal) sum of CM points in X of CM type (K, Φ) by OK . Then CM(K) = CM(K, Φ)+CM(K, Φ′) is an 0-cycle on X defined over Q. If Ψ is a rational modular function on X, then Ψ(CM(K)) is a rational number. An interesting and in general very hard question is to find a factorization formula for this number. We did it successfully when Ψ is a Borcherds product or equivalently has its divisor supported on the Hirzebruch-Zagier divisors, which were constructed in their seminar work in 1970’s [HZ]. Let K be the reflex field of (K, Φ) with real quadratic subfield F . For a nonzero element t ∈ d−1 K/F (relative discriminant) and a prime ideal l of F , we define
76 citations
24 Apr 2006
TL;DR: The method can be used to attack two fast RSA variants recently proposed by Galbraith, Heneghan, McKee, and by Sun, Wu and also present a second result for balanced RSA primes in the case that the public exponent e is significantly smaller than N.
Abstract: It is well-known that there is an efficient method for decrypting/signing with RSA when the secret exponent d is small modulo p–1 and q–1. We call such an exponent d a small CRT-exponent. It is one of the major open problems in attacking RSA whether there exists a polynomial time attack for small CRT-exponents, i.e. a result that can be considered as an equivalent to the Wiener and Boneh-Durfee bound for small d. At Crypto 2002, May presented a partial solution in the case of an RSA modulus N=pq with unbalanced prime factors p and q. Based on Coppersmith's method, he showed that there is a polynomial time attack provided that q
67 citations
TL;DR: A survey of recent developments of the Hardy-Littlewood method in analytic number theory can be found in this paper, where the author and Tao give an asymptotic for the number of 4-term arithmetic progressions of primes less than N.
Abstract: The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov's 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the Hardy-Littlewood method has been generalised to obtain, for example, an asymptotic for the number of 4-term arithmetic progressions of primes less than N.
58 citations
1 Dec 2006
TL;DR: A panorama of techniques used by modern analytic number theoreticists in the study of prime numbers can be found in this paper, where the primes are captured by adopting new axioms for sieve theory.
Abstract: The classical memoir by Riemann on the zeta function was motivated by questions
about the distribution of prime numbers. But there are important problems concerning prime
numbers which cannot be addressed along these lines, for example the representation of primes
by polynomials. In this talk Iwill showa panorama of techniques, which modern analytic number
theorists use in the study of prime numbers. Among these are sieve methods. I will explain how
the primes are captured by adopting new axioms for sieve theory. I shall also discuss recent
progress in traditional questions about primes, such as small gaps, and fundamental ones such
as equidistribution in arithmetic progressions. However, my primary objective is to indicate the
current directions in Prime Number Theory.
53 citations
Book•
1 Oct 2006
TL;DR: In this paper, the authors introduce primality testing for algebraic number theory, and present an overview of the primality test and its application to algebraic numbers and their applications in number theory.
Abstract: and Historical Remarks.- Basic Number Theory.- The Infinitude of Primes.- The Density of Primes.- Primality Testing: An Overview.- Primes and Algebraic Number Theory.
49 citations
1 Apr 2006
TL;DR: Goldston, Pintz, and Yoldorom as mentioned in this paper showed that an improvement of the Bombieri-Vinogradov prime number theorem would give rise infinitely often to bounded differences between consecutive primes.
Abstract: In the recent preprint (3), Goldston, Pintz, and Yoldorom established, among other things, (0) liminf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in (3). While (3) also includes quantitative versions of (0), we are concerned here solely with proving the qualitative (0), which still exhibits all the essentials of the method. We also show here that an improvement of the Bombieri-Vinogradov prime number theorem would give rise infinitely often to bounded differences between consecutive primes. We include a short expository last section. Detailed discussions of quantitative results and a historical review will appear in the publication version of (3) and its continuations.
47 citations
Posted Content•
TL;DR: In this paper, an expository article on the recent marvellous theorem of Goldston, Pintz, and Yildirim on small gaps between prime numbers is presented.
Abstract: This is an expository article on the recent marvellous theorem of Goldston, Pintz, and Yildirim on small gaps between prime numbers.
46 citations
Posted Content•
TL;DR: For any measure-preserving system, this article showed that there are infinitely many primes $p$ such that the von Mangoldt function can be expressed in terms of a common difference of the form $p-1$ (or $p+1$) for some prime $p.
Abstract: For any measure preserving system $(X,\mathcal{X},\mu,T)$ and $A\in\mathcal{X}$ with $\mu(A)>0$, we show that there exist infinitely many primes $p$ such that $\mu\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$ (the same holds with $p-1$ replaced by $p+1$). Furthermore, we show the existence of the limit in $L^2(\mu)$ of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form $p-1$ (or $p+1$) for some prime $p$.
TL;DR: In this article, the authors remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.
Abstract: We remove logarithmic factors in error term estimates in asymptotic formulas for the number of solutions of a class of additive congruences modulo a prime number.
1 Jan 2006
TL;DR: The p-adic tame level 1 eigencurve introduced by Coleman-Mazur is smooth at the evil Eisenstein points and necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of padic zeta values are given in this paper.
Abstract: Let p be a prime number and C be the p-adic tame level 1 eigencurve introduced by Coleman-Mazur. We prove that C is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of p-adic zeta values. A key step is the determination at these points of the schematic reducibility locus of the pseudo-character carried by C restricted to a decomposition group at p. Then, the smoothness appears to be a consequence of the fact that the Dirichlet L-functions only have simple zeros at integers.
Posted Content•
TL;DR: The existence of nonlinear balanced symmetric polynomials over GF(p) was shown in this paper, where the authors showed that for GF(n,p), there exists a polynomial-time nonlinear symmetric elementary symmetric over GF (2), where n is a prime number.
Abstract: Under mild conditions on $n,p$, we give a lower bound on the number of $n$-variable balanced symmetric polynomials over finite fields $GF(p)$, where $p$ is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that $X(2^t,2^{t+1}l-1)$ are the only nonlinear balanced elementary symmetric polynomials over GF(2), where $X(d,n)=\sum_{i_1
1 Jan 2006
TL;DR: In this paper, a generalization of Euclid's elementary proof of the infinitude of primes can be found for algebraic number fields, where the residue class (mod $k$) has order 1 or 2.
Abstract: We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod $k$) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.
1 Jan 2006
TL;DR: Recently, Tao as discussed by the authors showed that the Hardy-Littlewood method can be generalized to obtain an asymptotic for the number of 4-term arithmetic progressions of primes less than N.
Abstract: The Hardy�Littlewood method is a well-known technique in analytic number theory.
Among its spectacular applications are Vinogradov�s 1937 result that every sufficiently large
odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving
an asymptotic for the number of 3-term progressions of primes, all less than N. This article
surveys recent developments of the author and T. Tao, in which the Hardy�Littlewood method
has been generalised to obtain, for example, an asymptotic for the number of 4-term arithmetic
progressions of primes less than N.
TL;DR: In this paper, Ribet and Wiles gave new results about the description of the primitive solutions of the diophantine equation, in case the product of the prime divisors of divides with an odd prime number.
Abstract: Let be a prime number ≥ 5 and be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation , in case the product of the prime divisors of divides , with an odd prime number. For instance, under some conditions on , we provide a constant such that there are no such solutions if . In application, we obtain information concerning the -rational points of hyperelliptic curves given by the equation with .
TL;DR: It is proved that for every n there is a unique regular triangular embedding of K"n"," n","n"n in an orientable surface.
TL;DR: In this article, the congruences modulo the primary numbers n =p were studied for the traces of the integer matrices A and A n-φ(n), where A is an integer matrix and φ n is the number of residues modulo n, relatively prime to n.
Abstract: The congruences modulo the primary numbers n=p
a
are studied for the traces of the matrices A
n
and A
n-φ(n), where A is an integer matrix and φ(n) is the number of residues modulo n, relatively prime to n. We present an algorithm to decide whether these congruences hold for all the integer matrices A, when the prime number p is fixed. The algorithm is explicitly applied for many values of p, and the congruences are thus proved, for instance, for all the primes p ≤ 7 (being untrue for the non-primary modulus n=6). We prove many auxiliary congruences and formulate many conjectures and problems, which can be used independently.
23 Jul 2006
TL;DR: A new algorithm for computing the p-primary part of the ideal class group of F using Gauss sums and cyclotomic units is proposed.
Abstract: For an abelian number field F and an odd prime number p which does not divide the degree [F:ℚ], we propose a new algorithm for computing the p-primary part of the ideal class group of F using Gauss sums and cyclotomic units.
TL;DR: In this article, it was shown that the number of distinct prime divisors of an elliptic curve is ω( # E ( F p ) − log log p log log ω log p ω ω ( # E( F p ), where ω p is the exponent of E(F p ).
TL;DR: Fan et al. as mentioned in this paper considered p-adic affine dynamical systems on the ring Z p of all padic integers, and they found a necessary and sufficient condition for such a system to be minimal.
23 Jul 2006
TL;DR: Couveignes, Edixhoven and de Jong as discussed by the authors showed that the mod l Galois representation associated to the discriminant modular form Δ can be computed in time polynomial in l.
Abstract: We give an overview of the recent result by Jean-Marc Couveignes, Bas Edixhoven and Robin de Jong that says that for l prime the mod l Galois representation associated to the discriminant modular form Δ can be computed in time polynomial in l. As a consequence, Ramanujan’s τ(p) for prime numbers p can be computed in time polynomial in logp.
The mod l Galois representation occurs in the Jacobian of the modular curve X1(l), whose genus grows quadratically with l. The challenge therefore is to do the necessary computations in time polynomial in the dimension of this Jacobian. The field of definition of the l2 torsion points of which the representation consists is found via a height estimate, obtained from Arakelov theory, combined with numerical approximation. The height estimate implies that the required precision for the approximation grows at most polynomially in l.
Posted Content•
TL;DR: In this paper, it was shown that every element of a fully ramified, elementary abelian extension of a prime number with valuation congruent mod $[N:K]$ to the largest lower ramification number of $N/K$ generates a normal basis for any element of the extension.
Abstract: Let $p$ be a prime number and let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $N$ be a fully ramified, elementary abelian extension of $K$. Under a mild hypothesis on the extension $N/K$, we show that every element of $N$ with valuation congruent mod $[N:K]$ to the largest lower ramification number of $N/K$ generates a normal basis for $N$ over $K$.
TL;DR: In this article, the Stickelberger ideal of the group ring Z [ Δ ] was studied and the Hilbert-Speiser type property for the rings of pintegers of elementary abelian extensions over F of exponent p was checked.
Abstract: Let p be an odd prime number and F a number field. Let K = F ( ζ p ) and Δ = G a l ( K / F ) . Let 𝒮 Δ be the Stickelberger ideal of the group ring Z [ Δ ] defined in the previous paper [8]. As a consequence of a p -integer version of a theorem of McCulloh [15], [16], it follows that F has the Hilbert-Speiser type property for the rings of p -integers of elementary abelian extensions over F of exponent p if and only if the ideal 𝒮 Δ annihilates the p -ideal class group of K . In this paper, we study some properties of the ideal 𝒮 Δ ,and check whether or not a subfield of Q ( ζ p ) satisfies the above property.
TL;DR: In this paper, under the assumption of the Generalized Riemann Hypothesis, the natural densities of a positive integer which is not a perfect b -th power with b ≥ 2, q be a prime number and q a (x ; q i, j ) be the set of primes p ≤ x such that the residual order of a ( m o d p ) in ( Z / p Z ) × is congruent to q modulo q i.
Abstract: Let a be a positive integer which is not a perfect b -th power with b ≥ 2 , q be a prime number and Q a ( x ; q i , j ) be the set of primes p ≤ x such that the residual order of a ( m o d p ) in ( Z / p Z ) × is congruent to j modulo q i . In this paper, which is a sequel of our previous papers [1] and [6], under the assumption of the Generalized Riemann Hypothesis, we determine the natural densities of Q a ( x ; q i , j ) for i ≥ 3 if q = 2 , i ≥ 1 if q is an odd prime, and for an arbitrary nonzero integer j (the main results of this paper are announced without proof in [3], [7] and [2]).
27 Dec 2006
TL;DR: In this survey, three algorithms for testing primality of numbers that use Fermat's Little Theorem are described.
Abstract: In this survey, we describe three algorithms for testing primality of numbers that use Fermat's Little Theorem.
TL;DR: In this paper, a simplified version of Voevodsky's proof of the Milnor-Kato conjecture is presented, which does not need to consider elds transcendental over F and makes no use of motivic cohomology at all.
Abstract: m ) from the Milnor K-theory of the eld F modulo m to the Galois cohomology of F with cyclotomic coecien ts is an isomorphism (here, as usually, m denotes the group of m-roots of unity in F ). One can see that it suces to verify this conjecture in the case when m is a prime number. Several years ago V. Voevodsky outlined his approach to the Milnor{Kato conjecture in the preprint [10]. The rst step of his argument dealt with a prime number ‘, a eld F having no nite extensions of degree prime to ‘, and an integer n > 1 such that the group K M n+1 (F ) is ‘-divisible. Assuming that the conjecture holds in degree less or equal to n for m = ‘ and any eld containing F , Voevodsky was proving that H n+1 (GF; Z=‘) = 0. The main goal of this paper is to give a simplied elementary version of Voevodsky’s proof of this step. In particular, we do not need to consider elds transcendental over F and we make no use of motivic cohomology at all. Our main result is formulated as follows.
TL;DR: In this paper, a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors, are studied and the trace of the transfer matrix is shown to depend on the number of free parameters.
Abstract: Our starting point is a class of braid matrices, presented in a previous paper, constructed on a basis of a nested sequence of projectors. Statistical models associated to such $N^2\times N^2$ matrices for odd $N$ are studied here. Presence of $\frac 12(N+3)(N-1)$ free parameters is the crucial feature of our models, setting them apart from other well-known ones. There are $N$ possible states at each site. The trace of the transfer matrix is shown to depend on $\frac 12(N-1)$ parameters. For order $r$, $N$ eigenvalues consitute the trace and the remaining $(N^r-N)$ eigenvalues involving the full range of parameters come in zero-sum multiplets formed by the $r$-th roots of unity, or lower dimensional multiplets corresponding to factors of the order $r$ when $r$ is not a prime number. The modulus of any eigenvalue is of the form $e^{\mu\theta}$, where $\mu$ is a linear combination of the free parameters, $\theta$ being the spectral parameter. For $r$ a prime number an amusing relation of the number of multiplets with a theorem of Fermat is pointed out. Chain Hamiltonians and potentials corresponding to factorizable $S$-matrices are constructed starting from our braid matrices. Perspectives are discussed.
TL;DR: In this paper, it was shown that the congruence of the traces of the sums of powers of an algebraic integer matrices for the case that the exponent of the power is a prime power has been proved.
Abstract: The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power.