Abstract: For elliptic curves over Q, we prove the p-indivisibility of derived Heegner points for certain prime numbers p, as conjectured by Kolyvagin in 1991. Applications include the rened Birch{SwinertonDyer conjecture in the analytic rank one case, and a converse to the theorem of Gross{Zagier and Kolyvagin. A slightly dierent version of the converse is also proved earlier by Skinner.

Abstract: We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, obtaining an exponent of distribution $\frac{1}{2} + \frac{7}{300}$

Abstract: An improved method for data encryption has been developed. The method includes storing data, multiple prime numbers and random numbers within an electronic memory storage device. Next, calculating a public number using the multiple prime numbers and providing a public number to a recipient apparatus that has knowledge of the multiple prime numbers. The method then encrypts the stored data with a randomly generated key and deletes the randomly generated key after use. Next, the method calculates a common shared secret between the sender and recipient using the prime numbers, a recipient public number and the second random number. The sender and recipient calculate parameters using a key equation based on the randomly generated key and random numbers and a common shared secret. Finally, the recipient calculates the randomly generated key for decryption using the common shared secret, one of the prime numbers, the parameters and the simultaneous equations for decryption of the data.

Abstract: A classification up to isomorphism of all left braces of order $p^3$, where $p$ is any prime number, is given. To this end, we first classify all the left braces of order $p$ and $p^2$, and then we construct explicitly the hypothesis required in Corollary D of N. Ben David's Ph.D. thesis to build multiplications of left braces.

Abstract: Let $K$ be a global field which contains a primitive $p$-th root of unity, where $p$ is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for $p=2$, any triple Massey product over $K$ with respect to $\mathbb{F}_p$, contains 0 whenever it is defined. We show that this is true for all primes $p$.

Abstract: The Twin Prime conjecture states that there are infinitely many pairs of distinct primes which differ by $2$. Until recently this conjecture had seemed to be far out of reach with current techniques. However, in April 2013, Yitang Zhang proved the existence of a finite bound $B$ such that there are infinitely many pairs of distinct primes which differ by no more than $B$. This is a massive breakthrough, making the twin prime conjecture look highly plausible, and the techniques developed help us to better understand other delicate questions about prime numbers that had previously seemed intractable. Zhang even showed that one can take $B = 70000000$. Moreover, a co-operative team, \emph{polymath8}, collaborating only on-line, had been able to lower the value of $B$ to ${4680}$. They had not only been more careful in several difficult arguments in Zhang's original paper, they had also developed Zhang's techniques to be both more powerful and to allow a much simpler proof (and forms the basis for the proof presented herein). In November 2013, inspired by Zhang's extraordinary breakthrough, James Maynard dramatically slashed this bound to $600$, by a substantially easier method. Both Maynard, and Terry Tao who had independently developed the same idea, were able to extend their proofs to show that for any given integer $m\geq 1$ there exists a bound $B_m$ such that there are infinitely many intervals of length $B_m$ containing at least $m$ distinct primes. We will also prove this much stronger result herein, even showing that one can take $B_m=e^{8m+5}$.

Abstract: We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2n, where n is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.

Abstract: Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this Prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the Prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the Prime state, which in turn produces new insights in the Hardy-Littlewood conjecture for the pairwise distribution of primes. The extension of these ideas to a Twin Prime state shows that this new state is even more entangled than the Prime state, obeying majorization relations. We further discuss the construction of quantum states that encompass relevant series of numbers and opens the possibility of applying quantum computation to Arithmetics in novel ways.

Abstract: We shall prove an unconditional basic result related to the value- distributions of {(L′∕L)(s, χ)} χ and of {(ζ′∕ζ)(s + iτ)} τ , where χ runs over Dirichlet characters with prime conductors and τ runs over R. The result asserts that the expected density function common for these distributions are in fact the density function in an appropriate sense. Under the generalized Riemann hypothesis, stronger results have been proved in our previous articles, but our present result is unconditional.

Abstract: The ternary Goldbach conjecture, or three-primes problem, states that every odd number n greater than 5 can be written as the sum of three primes The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in the twentieth century In 2013 - following a line of research pioneered and developed by Hardy, Littlewood and Vinogradov, among others - the author proved the conjecture In this, as in many other additive problems, what is at issue is really the proper usage of the limited information we possess on the distribution of prime numbers The problem serves as a test and whetting-stone for techniques in analysis and number theory - and also as an incentive to think about the relations between existing techniques with greater clarity We will go over the main ideas of the proof The basic approach is based on the circle method, the large sieve and exponential sums For the purposes of this overview, we will not need to work with explicit constants; however, we will discuss what makes certain strategies and procedures not just effective, but efficient, in the sense of leading to good constants Still, our focus will be on qualitative improvements Mathematics Subject Classification (2010) Primary 11P32

Abstract: We prove that for n>2 there exists a quandle of cyclic type of size n if and only if n is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of cyclic type. This establishes a conjecture of H. Tamaru.

Abstract: In this paper, we found that (p,q,x,y,z) = (3,5,1,0,2) is a unique solution of the Diophantine equation p x + q y = z 2 where p is an odd prime number which q − p = 2 and x, y and z are non-negative integers.

Abstract: Let $\Lambda(n)$ be the von Mangoldt function, $x$ real and $2\leq y \leq x$. This paper improves the estimate on the exponential sum over primes in short intervals \[ S_k(x,y;\alpha) = \sum_{x< n \leq x+y} \Lambda(n) e\left( n^k \alpha \right) \] when $k\geq 3$ for $\alpha$ in the minor arcs. And then combined with the Hardy--Littlewood circle method, this enables us to investigate the Waring--Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers, which improves the results in Wei and Wooley [12].

Abstract: The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.

Abstract: For an elliptic curve over the rational number field and a prime number p, we study the structure of the classical Selmer group of p-power torsion points In our previous paper Kurihara (Refined Iwasawa theory for p-adic representations and the structure of Selmer groups Munster J Math http://wwwmathkeioacjp/~kurihara/, to appear), assuming the main conjecture and the non-degeneracy of the p-adic height pairing, we proved that the structure of the Selmer group with respect to p-power torsion points is determined by some analytic elements \(\tilde{\delta }_{m}\) defined from modular symbols (see Theorem 1 below) In this paper, we do not assume the main conjecture nor the non-degeneracy of the p-adic height pairing, and study the structure of Selmer groups (see Theorems 3 and 4), using these analytic elements and Kolyvagin systems of Gauss sum type

Abstract: We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sarkozy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.

Abstract: We give non-torsion counterexamples against the integral Tate conjecture for finite fields. We extend the result due to Pirutka and Yagita for prime numbers 2,3,5 to all prime numbers.

Abstract: Using the Poincare–Birkhoff fixed point theorem, we prove that for every β > 0 and for a large (both in the sense of prevalence and of category) set of continuous and T-periodic functions $${f: \mathbb{R} \to \mathbb{R}}$$ with $${\int_0^T f(t)\,dt = 0}$$ , the forced pendulum equation $$x'' + \beta \sin x = f(t) $$ has a subharmonic solution of order k for every large integer number k. This improves the well known result obtained with variational methods, where the existence when k is a (large) prime number is ensured.

Abstract: Chebyshev observed in a letter to Fuss that there tends to be more primes of the form 4n+3 than of the form 4n+1. The general phenomenon, which is referred to as Chebyshev’s bias, is that primes tend to be biased in their distribution among the different residue classes modq. It is known that this phenomenon has a strong relation with the low-lying zeros of the associated L-functions, that is, if these ‘L‘-functions have zeros close to the real line, then it will result in a lower bias. According to this principle one might believe that the most biased prime number race we will ever find is the Li(x) versus π(x) race, since the Riemann zeta function is the ‘L‘-function of rank one having the highest first zero. This race has density 0.99999973…, and we study the question of whether this is the highest possible density. We will show that it is not the case; in fact, there exist prime number races whose density can be arbitrarily close to 1. An example of a race whose density exceeds the above number is the race between quadratic residues and nonresidues modulo 4849845, for which the density is 0.999999928…. We also give fairly general criteria to decide whether a prime number race is highly biased or not. Our main result depends on the generalized Riemann hypothesis and a hypothesis on the multiplicity of the zeros of a certain Dedekind zeta function. We also derive more precise results under a linear independence hypothesis.

Abstract: We establish a conditional equivalence between quantitative unboundedness of the analytic rank of elliptic curves over $\mathbb Q$ and the existence of highly biased elliptic curve prime number races. We show that conditionally on a Riemann Hypothesis and on a hypothesis on the multiplicity of the zeros of $L(E,s)$, large analytic ranks translate into an extreme Chebyshev bias. Conversely, we show under a certain linear independence hypothesis on zeros of $L(E,s)$ that if highly biased elliptic curve prime number races do exist, then the Riemann Hypothesis holds for infinitely many elliptic curve $L$-functions and there exist elliptic curves of arbitrarily large rank.

Abstract: In this paper, we study representations of the algebra \(\mathcal {A}\) generated by all arithmetic functions, determined by fixed primes (or prime numbers). The main purposes of this paper are (i) to establish nice representational models of \(\mathcal {A}\) under primes, (ii) to study fundamental properties of such representations, (iii) to investigate how \( \mathcal {A}\) is acting as operators in representations, and (iv) to consider new free probability models on Krein-space operator algebras.

Abstract: We propose various strategies for improving the computation of discrete logarithms in non-prime fields of medium to large characteristic using the Number Field Sieve. This includes new methods for selecting the polynomials; the use of explicit automorphisms; explicit computations in the number fields; and prediction that some units have a zero virtual logarithm. On the theoretical side, we obtain a new complexity bound of $L_{p^n}(1/3,\sqrt[3]{96/9})$ in the medium characteristic case. On the practical side, we computed discrete logarithms in $F_{p^2}$ for a prime number $p$ with $80$ decimal digits.

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