Abstract: Let $$a_0\in \{0,\ldots ,9\}$$ . We show there are infinitely many prime numbers which do not have the digit $$a_0$$ in their decimal expansion. The proof is an application of the Hardy–Littlewood circle method to a binary problem, and rests on obtaining suitable ‘Type I’ and ‘Type II’ arithmetic information for use in Harman’s sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates rely on a combination of the geometry of numbers, the large sieve and moment estimates obtained by comparison with a Markov process.

Abstract: In this paper, we show an equi-disctributed property in $2$-dimensional finite abelian groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ where $p$ is a prime number. By using this equi-disctributed property, we prove that Fuglede's spectral set conjecture holds on groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$, namely, a set in $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ is a spectral set if and only if it is a tile.

Abstract: The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

Abstract: We show that for any prime odd integer $d$, there exists a correlation of size $\Theta(r)$ that can robustly self-test a maximally entangled state of dimension $4d-4$, where $r$ is the smallest multiplicative generator of $\mathbb{Z}_d^\ast$. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. Vol. $7$, ($2019$)). Since there are infinitely many prime numbers whose smallest multiplicative generator is at most $5$ (M. Murty The Mathematical Intelligencer $10.4$ ($1988$)), our result implies that constant-sized correlations are sufficient for robust self-testing of maximally entangled states with unbounded local dimension.

Abstract: Maximal connected grading classes of a finite-dimensional algebra $A$ are in one-to-one correspondence with Galois covering classes of $A$ which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental group $\pi_1(A)$. Our first concern here is the algebras $A=M_n(\mathbb{C})$. Their maximal connected gradings turn out to be in one-to-one correspondence with the Aut$(G)$-orbits of non-degenerate classes in $H^2(G,\C^*)$, where $G$ runs over all groups of central type whose orders divide $n^2$. We show that there exist groups of central type $G$ such that $H^2(G,\C^*)$ admits more than one such orbit of non-degenerate classes. We compute the family $\Lambda$ of positive integers $n$ such that there is a unique group of central type of order $n^2$, namely $C_n\times C_n$. The family $\Lambda$ is of square-free integers and contains all prime numbers. It is obtained by a full description of all groups of central type whose orders are cube-free. We establish the maximal connected gradings of all finite dimensional semisimple complex algebras using the fact that such gradings are determined by dimensions of complex projective representations of finite groups. In some cases we give a description of the corresponding fundamental groups.

Abstract: We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL 2[Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of ex...

Abstract: Let l be a prime number. We show that the Morita Frobenius number of an l-block of a quasi-simple finite group is at most 4 and that the strong Frobenius number is at most 4jDj2!, where D denotes a defect group of the block. We deduce that a basic algebra of any block of the group algebra of a quasi-simple finite group over an algebraically closed field of characteristic l is defined over a field with la elements for some a ≤ 4. We derive consequences for Donovan's conjecture. In particular, we show that Donovan's conjecture holds for l-blocks of special linear groups.

Abstract: The zeta function ζ ( s ) today is the oldest and most important tool to study the distribution of prime numbers and is the simplest example of a whole class of similar functions, equally important for understanding the deepest problems of number theory. The celebrated Riemann hypothesis is that all complex zeros of ζ ( s ) have real part equal to 1 2 . The consequences of a proof and even of an unlikely disproof of this hypothesis would be a giant step forward for understanding prime numbers. The paper by Griffin et al. (1) makes fundamental progress in the study of the Riemann zeta function by introducing a method to study certain classical polynomials (the so-called Jensen polynomials) that were known to play a role for understanding the finer properties of the zeta function but had proved to be quite intractable to study by means of standard methods. What was known before this work was a plausible but inaccessible conjecture, called hyperbolicity, for all of them. In this paper the authors introduce a method to study these polynomials which allows the authors to … [↵][1]1Email: eb{at}math.ias.edu. [1]: #xref-corresp-1-1

Abstract: The primary goal of this paper is to investigate the geometry of the $p$-adic eigencurve at a point $f$ corresponding to a weight one cuspidal theta series irregular at the prime number $p$. We show that $f$ belongs to exactly three or four irreducible components and study their intersection multiplicities. In particular, we show that the congruence ideal of a CM component has a simple zero at $f$ if and only if a certain anti-cyclotomic $\mathscr{L}$-invariant $\mathscr{L}_-(\varphi)$ does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst $\mathscr{L}_-(\varphi)$ and $\mathscr{L}_-(\varphi^{-1})$ is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz $p$-adic $L$-function of $\varphi$ has a simple (trivial) zero at $s=0$ if $\mathscr{L}_-(\varphi)$ is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz $p$-adic $L$-function of $\varphi$ at $s=0$ extending a conjecture of Gross.

Abstract: In this work, we retake an old idea that Koblitz presented in his landmark paper (Koblitz, in: Proceedings of CRYPTO 1991. LNCS, vol 576, Springer, Berlin, pp 279–287, 1991), where he suggested the possibility of defining anomalous elliptic curves over the base field $${\mathbb {F}}_4$$ . We present a careful implementation of the base and quadratic field arithmetic required for computing the scalar multiplication operation in such curves. We also introduce two ordinary Koblitz-like elliptic curves defined over $${\mathbb {F}}_4$$ that are equipped with efficient endomorphisms. To the best of our knowledge, these endomorphisms have not been reported before. In order to achieve a fast reduction procedure, we adopted a redundant trinomial strategy that embeds elements of the field $${\mathbb {F}}_{4^{m}},$$ with m a prime number, into a ring of higher order defined by an almost irreducible trinomial. We also suggest a number of techniques that allow us to take full advantage of the native vector instructions of high-end microprocessors. Our software library achieves the fastest timings reported for the computation of the timing-protected scalar multiplication on Koblitz curves, and competitive timings with respect to the speed records established recently in the computation of the scalar multiplication over binary and prime fields.

Abstract: Large integer factorization is one of the basic issues in number theory and is the subject of this paper. Our research shows that the Pisano period of the product of two prime numbers (or an integer multiple of it) can be derived from the two prime numbers themselves and their product, and we can therefore decompose the two prime numbers by means of the Pisano period of their product. We reduce the computational complexity of modulo operation through the “fast Fibonacci modulo algorithm” and design a stochastic algorithm for finding the Pisano periods of large integers. The Pisano period factorization method, which is proved to be slightly better than the quadratic sieve method and the elliptic curve method, consumes as much time as Fermat's method, the continued fractional factorization method and the Pollard p-1 method on small integer factorization cases. When factoring super-large integers, the Pisano period factorization method has shown as strong performance as subexponential complexity methods; thus, this method demonstrates a certain practicability. We suggest that this paper may provide a completely new idea in the area of integer factorization problems.

Abstract: Exploiting the equidistribution properties of polynomial sequences, following the methods developed by Leibman ("Pointwise Convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam. Systems, 25 (2005) no. 1, 201-213") and Frantzikinakis ("Multiple recurrence and convergence for Hardy field sequences of polynomial growth. Journal d'Analyse Mathematique, 112 (2010), 79-135" and "Equidistribution of sparse sequences on nilmanifolds. Journal d'Analyse Mathematique, 109 (2009), 353-395") we show that the ergodic averages with iterates given by the integer part of real-valued strongly independent polynomials, converge in the mean to the "right"-expected limit. These results have, via Furstenberg's correspondence principle, immediate combinatorial applications while combining these results with methods from "The polynomial multidimensional Szemeredi theorem along shifted primes. Israel J. Math., 194 (2013), no. 1, 331-348" and "Closest integer polynomial multiple recurrence along shifted primes. Ergodic Theory Dynam. Systems, 1-20. doi:10.1017/etds.2016.40" we get the respective "right" limits and combinatorial results for multiple averages for a single sequence as well as for several sequences along prime numbers.

Abstract: Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + l for | l | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.

Abstract: We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by using the classification of finite simple groups.

Abstract: Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is #GraphHom(H) --- the problem of finding the number of homomorphisms from a given graph G to the graph H. Not only the complexity of this basic problem is known, but also of its many variants for digraphs, more general relational structures, graphs with weights, and others. In this paper we consider a modification of #GraphHom(H), the #_p GraphHom(H) problem, p a prime number: Given a graph G, find the number of homomorphisms from G to H modulo p. In a series of papers Faben and Jerrum, and Goebel et al. determined the complexity of #_2 GraphHom(H) in the case H (or, in fact, a certain graph derived from H) is square-free, that is, does not contain a 4-cycle. Also, Goebel et al. found the complexity of #_p GraphHom(H) for an arbitrary prime p when H is a tree. Here we extend the above result to show that the #_p GraphHom(H) problem is #_p P-hard whenever the derived graph associated with H is square-free and is not a star, which completely classifies the complexity of #_p GraphHom(H) for square-free graphs H.

Abstract: In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$ , the basic idea is to fix a constant $q\propto x^{1- \varepsilon }$ , pick a uniformly random $a coprime to $q$ , and choose $p$ of the form $a+t\cdot q$ , where only $t$ is updated if the primality test fails. We prove that variants of this approach provide prime generation algorithms requiring a few random bits and whose output distribution is close to uniform, under less and less expensive assumptions: first a relatively strong conjecture by H. Montgomery, made precise by Friedlander and Granville; then the Extended Riemann Hypothesis; and finally fully unconditionally using the Barban–Davenport–Halberstam theorem. We argue that this approach has a number of desirable properties compared with the previous algorithms, at least in an asymptotic sense. In particular: 1) it uses much fewer random bits than both the “trivial algorithm” (testing random numbers less than $x$ for primality) and Maurer’s almost uniform prime generation algorithm; 2) the distance of its output distribution to uniform can be made arbitrarily small, unlike algorithms like PRIMEINC (studied by Brandt and Damgard), which we show exhibit significant biases; and 3) all quality measures (number of primality tests, output entropy, randomness, and so on) can be obtained under standard conjectures or even unconditionally, whereas most previous nontrivial algorithms can only be proved based on stronger, less standard assumptions like the Hardy-Littlewood prime tuple conjecture. Note, however, that our analysis involves non-explicit constants, and therefore does not establish the superiority of our approach for concrete parameter sizes.

Abstract: Let be a prime number. We use a number field variant of Vinogradov’s method to prove density results about the following four arithmetic invariants: (i) -rank of the class group of the imaginary quadratic number field ; (ii) -rank of the ordinary class group of the real quadratic field ; (iii) the solvability of the negative Pell equation over the integers; (iv) -part of the Tate–Safarevic group of the congruent number elliptic curve . Our results are conditional on a standard conjecture about short character sums.

Abstract: This chapter presents the RSA cryptosystem, a much-heralded cryptosystem used worldwide. Large prime numbers are needed to construct an RSA cryptosystem, so the second half of the chapter is devoted to seeing how many large primes there are, and how to identify large primes with high confidence.

Abstract: In this paper we establish a general asymptotic formula for the sum of the first n prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin in 1996.