Exponential sums and prime divisors of sparse integers

We obtain a new lower bound on the number of prime divisors of integers whose g-ary expansion contains a fixed number of nonzero digits.      

Prime divisors of palindromes

Summary In this paper, we study some divisibility properties of palindromic numbers in a fixed base <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>g\ge 2$. In particular, if ${\mathcal P}_L$ denotes the set of palindromes with precisely $L$ digits, we show that for any sufficiently large value of $L$ there exists a palindrome $n\in{\mathcal P}_L$ with at least $(\log\log n)^{1+o(1)}$ distinct prime divisors, and there exists a palindrome $n\in{\mathcal P}_L$ with a prime factor of size at least $(\log n)^{2+o(1)}$.     

A remark on a paper of Luca

Summary It is proved that the set of those natural numbers which cannot be written as n-Ω(n) is of positive lower density. Here Ω(n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca.     

Additive problems for integers with a given number of prime divisors

The asymptotics of sums of the form Στ(|bn−a|) (summation overn<N, ω(n)=k) is studied, whereω(n) is the number of distinct prime divisors ofn, andτ(n) is the number of all divisors. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 749–762, May, 1998. In conclusion, the author wishes to express his gratitude to Professor N. M. Timofeev for valuable advice. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00502.     

On Farey Fractions with Small Prime Factors

We investigate the Farey fractions, i.e., the set of irreducible fractions m /n, 0 < m < n x. We derive an asymptotic equality for the number of Farey fractions having no large prime factors.     

Primitive divisors of some Lehmer-Pierce sequences

We study primitive prime divisors of the terms of Δ(u)=(Δn(u))_n?1, where Δn(u)=NK/Q(un−1) for K a real quadratic field, and u a unit element of its ring of integers. The methods used allow us to find the terms of the sequence that do not have a primitive prime divisor.     

On sums and products of integers

Erdos and Szemerédi conjectured that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of . Erdos and Szemerédi proved that this number must be at least for some and . In this paper it is proved that the result holds for .

     

On sums and products of integers

Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

关于素变数丢番图方程组与不等式组（英文）

本文研究了和素数有关的方程组和不等式组，通过更细致的指数和估计，我们改进了以前的结果。     

关于Euler e-函数和n的指数互素因子

对于任意正整数n,设n=pα11pα22…pαrr为n的标准素因数分解式,如果对于de n且de=pβ11pβ22…pβrr有(βi,αi)=1(i=1,2,…,k),则称de为n的指数互素因子.本文利用初等及解析方法研究了正整数n的所有de因子的求和及求积的计算问题,获得了两个有趣的计算公式;同时还研究了n的所有de因子个数函数,即Eu ler e函-数φe(n)的均值性质,并给出了一个较强的渐近公式.     

Integers with dense divisors 2

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t).     

Properties of large prime divisors of numbers of the form p − 1

The main result of this paper is the fact that the fraction of primes p ≤ x satisfying the condition that p ? 1 has a prime divisor q > exp(ln x/ln ln x) and the number of prime divisors of q ? 1 essentially differ from ln ln(x/n), where n = (p ? 1)/q, tends to zero as x increases.     

Universally bad integers and the 2-adics

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if , where is the set of nonnegative integers whose 4-adic expansion has only 0's and 1's, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if (ua,b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2^k+1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u=φ_p^k(4) are universally bad, where p is prime and φ_n(x) is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u=φ_p^k(4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ_2^k(4), and more generally, those of the form 4^k+1, are not de Bruijn universally bad.     

On numbers not of the form <Emphasis Type=”Italic”>n</Emphasis>-ω(<Emphasis Type=”Italic”>n</Emphasis>)

Summary We introduce a new class of lightlike submanifolds, namely, Screen Cauchy Riemann (SCR) lightlike submanifolds of indefinite Kaehler manifolds. Contrary to CR-lightlike submanifolds, we show that SCR-lightlike submanifolds include invariant (complex) and screen real subcases of lightlike submanifolds. We study some properties of proper totally umbilical SCR-lightlike submanifolds, their invariant (complex) and screen real subcases.     

Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).     

Polynomial interpolation and sums of powers of integers

In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P_k(n) and Q_k(n), such that P_k(n) = Q_k(n) = f_k(n) for n = 1, 2,…?, k, where f_k(1), f_k(2),…?, f_k(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that f_k(n) is given by the sum of powers of the first n positive integers S_k(n) = 1^k + 2^k + ??? + n^k, and show that S_k(n) admits the polynomial representations S_k(n) = P_k(n) and S_k(n) = Q_k(n) for all n = 1, 2,…?, and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S_k(n) alternative to the well-known formula of Bernoulli.     

On sums of a prime and four prime squares in short intervals

In this paper, we prove that each sufficiently large integer N ≠1（mod 3） can be written as N=p＋p1^2＋p2^2＋p3^2＋p4^2, with
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes.     

GW invariants relative to normal crossing divisors

In this paper we introduce a notion of symplectic normal crossing divisor V and define the GW invariant of a symplectic manifold X relative to such a divisor. Our definition includes normal crossing divisors from algebraic geometry. The invariants we define in this paper are key ingredients in symplectic sum type formulas for GW invariants, and extend those defined in our previous joint work with T.H. Parker [16], which covered the case V   was smooth. The main step is the construction of a compact moduli space of relatively stable maps into the pair (X,V)$(X,V)$ in the case V is a symplectic normal crossing divisor in X.