Partitions into two Lehmer numbers

We prove an asymptotic formula for the number of representations of an integer as sum of two Lehmer numbers which implies that under some natural restrictions each sufficiently large integer is a sum of two such numbers.     

On some infinite sums of iterated radicals

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two particular sequences, a new characterization of this number will appear. In that process, an infinite sum of iterated radicals is obtained. Based on that result, this article will then proceed to analyse that sum. The conditions under which the infinite sum yields an integer will be inspected, thereby calculating the value of the sum. After that, a method is established to develop some algebraic irrationals as a continued fraction. Finally, the results will be applied to an infinite difference of iterated radicals.     

关于同余式$\sigma(n)\equiv 1\pmod n$, II

Let k ≥ 2 be an integer, and let a（n） denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if a（n） = kn ＋ 1. In this paper, we give some necessary properties of quasi-multiperfect numbers with four different prime divisors.     

On a Waring-Goldbach-type problem for fourth powers

In this paper, we prove that every sufficiently large positive integer satisfying some necessary congruence conditions can be represented by the sum of a fourth power of integer and twelve fourth powers of prime numbers.     

表整数为八个三角数的表法数目

通过留数定理把一个无究乘积展成Laurent级数,利用这个展式可以简单地证明表整数为八个三角数的表法数目公式。     

An algorithm for computing plethysm coefficients

The plethysm of two Schur functions can be expressed as a sum of Schur functions with nonnegative integer coefficients. Current algorithms for computing plethysms are designed to compute the whole expansion. However, in some applications only a few coefficients are of interest. In this work, we develop an algorithm for calculating individual plethysm coefficients. We also give a simple result concerning the zero coefficients which is obtained from the combinatorial properties of the Kostka and the inverse Kostka numbers.     

一个关于广义完全数的数论函数方程

本文运用初等方法,讨论了一个含有约数函数、约数和函数与Euler函数的数论函数方程,给出了该方程的全部偶数解,并且解决了一个有关广义完全数的问题.     

Melham's Conjecture on Odd Power Sums of Fibonacci Numbers

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at a certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at 1, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an a?rmative answer to a conjecture of Melham.     

On the Exponential Sum with Square-Free Numbers

A new estimate for the exponential sum with square-free numbersis established. This result is applied to the problem of findingthe number of representations of a large integer as a sum ofthree square-free numbers. 2000 Mathematics Subject Classification11L07, 11N36, 11P99.     

华蘅芳数在幂和问题中的新应用

自然效的幂和问题具有悠久的历史,亦不乏现代的兴趣.一般学者不了解清代数学家华蘅芳的成果.本文改进了华氏数的定义;针对该问题建立了新的取盒-放球模型,给出幂和的组合解释;应用华氏数获得了简捷的幂和公式.文末介绍了华氏数的历史来源.     

Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

Tight upper and lower bounds for the reciprocal sum of Proth primes

The Ramanujan Journal - Computing the reciprocal sum of sparse integer sequences with tight upper and lower bounds is far from trivial. In the case of Carmichael numbers or twin primes even the...     

On Finite Forms of Certain Bilateral Basic Hypergeometric Series and Their Applications

In this paper we derive finite forms of the summation formulas for bilateral basic hypergeometric series 3ψ3,4ψ4 and 5ψ5.We therefrom obtain the summation formulae obtained recently by Wenchang CHU and Xiaoxia WANG.As applications of these summation formulae,we deduce the well-known Jacobi's two and four square theorems,a formula for the number of representations of an integer n as sum of four triangular numbers and some theta function identities.     

The number of powers of 2 in a representation of large even integers (I)

Under the Generalized Riemann Hypothesis, it is proved that for any integer k⩾770 there is N_k>0 depending onk only such that every even integer ⩾N_k is a sum of two odd prime numbers andk powers of 2. The research is partially supported by RGC research grant (HKU 518/96P). The first author is supported by Post-Doctoral Fellowship of The University of Hong Kong.     

On the Waring-Goldbach Problem for Fourth and Fifth Powers

It is shown that every sufficiently large integer congruentto 14 modulo 240 may be written as the sum of 14 fourth powersof prime numbers, and that every sufficiently large odd integermay be written as the sum of 21 fifth powers of prime numbers.The respective implicit bounds 14 and 21 improve on the previousbounds 15 (following from work of Davenport) and 23 (due toThanigasalam). These conclusions are established through themedium of the Hardy-Littlewood method, the proofs being somewhatnovel in their use of estimates stemming directly from exponentialsums over prime numbers in combination with the linear sieve,rather than the conventional methods which ‘waste’a variable or two by throwing minor arc estimates down to anauxiliary mean value estimate based on variables not restrictedto be prime numbers. In the work on fifth powers, a switchingprinciple is applied to a cognate problem involving almost primesin order to obtain the desired conclusion involving prime numbersalone. 2000 Mathematics Subject Classification: 11P05, 11N36,11L15, 11P55.     

A naturally ordered enumeration of compositions

A composition of the positive integer M, with N-parts, is a vector of N non-negative integer components the sum of which is M. The paper presents a transformation that “enumerates” (assigns serial numbers) the set of all possible compositions most densely so that their “natural” ordering is preserved.     

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.     

Waring-Goldbach Problem: One Square and Nine Biquadrates

In this paper it is proved that every sufficiently large even integer N satisfying one of the congruence conditions N ≡ 10, 58, 130, or 178(mod 240) may be represented as the sum of one square and nine fourth powers of prime numbers.     

Divisibity,iterated digit sums,primality tests

The divisibility of numbers is obtained by iteration of the weighted sum of their integer digits. Then evaluation of the related congruences yields information about the primality of numbers in certain recursive sequences. From the row elements in generalized Delannoy triangles, we can verify the primality of any constellation of numbers. When a number set is not a prime constellation, we can identify factors of their composite numbers. The constellation primality test is proven in all generality, and examples are given for twin primes, prime triplets, and Sophie Germain primes.      

具有四个素因子的奇亏完全数

设n为自然数,σ(n)表示n的所有正因子和函数.令d是n的真因子,若n满足σ(n)=2n-d,则称n为亏因子为d的亏完全数.本文给出了具有四个素因子的奇亏完全数的一些性质的刻画.