Prime numbers and the first digit phenomenon

The set of primes which have lead digit 1 does not have relative natural density in the prime numbers. However, Bombieri has shown that this set does have relative Zeta density equal to log₁₀ 2. This means that a prime chosen at random (w.r.t. the Zeta distribution) will have lead digit 1 with the determined probability. Here the question, Is this a special property of Zeta density or a more universal property of primes? is answered. It is shown that for any generalization of relative natural density (obeying a few basic assumptions) if a value is assigned to the relative density of primes of lead digit 1 then this value is always log₁₀ 2. Another density which does converge on this set is also exhibited. Additionally the relative densities of primes beginning with any specified string of digits are found.     

Probabilistic algorithm for testing primality

We present a practical probabilistic algorithm for testing large numbers of arbitrary form for primality. The algorithm has the feature that when it determines a number composite then the result is always true, but when it asserts that a number is prime there is a provably small probability of error. The algorithm was used to generate large numbers asserted to be primes of arbitrary and special forms, including very large numbers asserted to be twin primes. Theoretical foundations as well as details of implementation and experimental results are given.     

小区间内一个素数和三个素数的平方和问题

研究了把一个满足必要条件的自然数在小区间内分解成一个素数和三个素数平方和的问题,利用刘建亚和展涛处理扩大了的主区间的新方法,成功的缩短了小区间的长度.     

论筛法及其有关的若干应用——殆素数的分布问题

<正> 本文的宗旨在于证明作者在[1]内所提及的全部结果,现在将本文的强果详述于下:定理1.命 F(x)表一无固定素因子的 k 次既约整值多项式.命(?)此处 w 是适合下面不等式的最小正整数(?)则在叙列{F(x)}中存在无限多个不超过 n 个素数的乘积.例如存在无限多个 x,使 x~3+2的素因子个数(包括相同的与相异的)不多于4.与此相类似,有定理2.设 k 为一正整数,命 n 适合(1)及(2),则当 x 充分大时,区间 x相似文献   

Numbers in a given set with (or without) a large prime factor

This survey treats two connected questions in analytic number theory: given a set of natural numbers, one may seek numbers with large prime factors in the set. Alternatively, one searches for smooth numbers in the set. Many examples have been studied: the set of values of a polynomial, the set of integers in a short interval, the set of shifted primes p+a and so on. These are discussed at some length, with references to the literature.     

More congruences for the coefficients of quotients of Eisenstein series

Berndt and Yee (Acta Arith. 104 (2002) 297) recently proved congruences for the coefficients of certain quotients of Eisenstein series. In each case, they showed that an arithmetic progression of coefficients is identically zero modulo a small power of 3 or 7. The present paper extends these results by proving that there are infinite classes of odd primes for which the set of coefficients that are zero modulo an arbitrary prime power is a set of arithmetic density one. A new family of explicit congruences modulo arbitrary powers of 2 is also found.     

论筛法及其有关的若干应用(Ⅰ)——表大整数为殆素数之和

<正> 1.结果的陈述本文的宗旨在于证明作者在[1]内所提及的非条件结果及[2]内所提及的全部结果.为简单见,将下面的命题记为(a,b):每一充分大的偶数可表为两个大于1的整数 c_1与 c_2之和,c_1与 c_2的素因子个数(包合相同的与相异的)分別不超过 a 与 b.并不需要很复杂的数值计算,就能得到(3,3)及(a,b),(a+b≤5).用比较复杂的数值计算,我们得到了(2,3).另一点值得注意的是本文所用的方法完全是初等的,而А.И.Виноградов在证明(3,3)的过程中却引用了精深的 Riemann 一ζ画数论的结果.     

Subtleties in the Distribution of the Numbers of Points on Elliptic Curves Over a Finite Prime Field

Three questions concerning the distribution of the numbers ofpoints on elliptic curves over a finite prime field are considered.First, the previously published bounds for the distributionare tightened slightly. Within these bounds, there are wildfluctuations in the distribution, and some heuristics are discussed(supported by numerical evidence) which suggest that numbersof points with no large prime divisors are unusually prevalent.Finally, allowing the prime field to vary while fixing the fieldof fractions of the endomorphism ring of the curve, the orderof magnitude of the average order of the number of divisorsof the number of points is determined, subject to assumptionsabout primes in quadratic progressions. There are implications for factoring integers by Lenstra's ellipticcurve method. The heuristics suggest that (i) the subtletiesin the distribution actually favour the elliptic curve method,and (ii) this gain is transient, dying away as the factors tobe found tend to infinity.     

表大偶數為一個不超過三個素數的乘積及一個不超過四個素數的乘積之和

<正> V.Brun最初在1920年證明了:每一充分大的偶數可表為兩個各不超過9個素數的乘積之和.簡記之為(9,9).後來,不少數學家改進與簡化了Brun方法,因此,Brun的結果也得到相應的改進,     

Two results on powers of 2 in Waring-Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.     

Primes in arithmetic progressions with friable indices

We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.     

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.     

关于调和数的一个结论

设n是大于1的正常数,并且设n=pα11p2α2…ptαt,其中pi为素数,i=1,2,…,t,ω(n)表示n的不同素因子的个数,即ω(n)=t.若n的所有因子的倒数和为整数,即0≤∑ij≤αjj=1,2,…,t1p1i1pi22…ptit为整数,称n是调和数.证明了和调和数相关的一个结论.     

表大偶數為一個素數及一個不超過四個素數的乘積之和——廣義Riemann猜測下之結果

<正> 引言 本文之目的是在證明作者在[1]內所提及的若干結果,本文所有的結果,均在廣義的Riemann猜測之下,而獲得的. 現在,先將廣義的Riemann猜測述於下:     

Some Applications of a Bombieri-Type Theoremin Totally Real Number Fields

 The primary concern of this paper is to present three further applications of a multi-dimensional version of Bombieri’s theorem on primes in arithmetic progressions in the setting of a totally real algebraic number field K. First, we deal with the order of magnitude of a greatest (relative to its norm) prime ideal factor of , where the product runs over prime arguments ω of a given irreducible polynomial F which lie in a certain lattice point region. Then, we turn our attention to the problem about the occurrence of algebraic primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. Finally, we give further contributions to the binary Goldbach problem in K. (Received 11 January 2000; in revised form 4 December 2000)     

Über die Verteilung von primen primitiven Wurzeln in algebraischen Zahlkörpern

Generalizing a method introduced by Elliott in the rational case to number fields in an appropriate way, asymptotic estimates are given for the number of algebraic primes in certain parallelotopes which are primitive roots for almost all (in a certain sense) prime ideal moduli. The proofs depend upon a fundamental lemma of Selberg's rational sieve method and make use of the large sieve in the setting of an algebraic number field.     

Some properties of elements of the prime field

We obtain some properties of primitive roots in the groups of residue class modulo prime number, in particular, Fermat primes and Sophie Germain primes. We also obtain some formulas for computation of products of elements in some subsets of the prime field of positive characteristic.     

Irreducibility of polynomials modulo p via Newton polytopes

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials.     

循环数域导子的一点注记

主要研究循环数域的导子公式.利用Kronecker-Weber定理及整体域的分歧理论,对于给定除子分歧个数的素数次循环扩域,明确给出了这类数域的导子公式及其个数.     

Ramanujan-Fourier series and the conjecture D of Hardy and Littlewood

We give a heuristic proof of a conjecture of Hardy and Littlewood concerning the density of prime pairs to which twin primes and Sophie Germain primes are special cases. The method uses the Ramanujan-Fourier series for a modified von Mangoldt function and the Wiener-Khintchine theorem for arithmetical functions. The failing of the heuristic proof is due to the lack of justification of interchange of certain limits. Experimental evidence using computer calculations is provided for the plausibility of the result. We have also shown that our argument can be extended to the m-tuple conjecture of Hardy and Littlewood.