Representations of integers as sums of powers with increasing exponents — II

This paper describes some computations and conjectures concerning the representation of integers as sums of powers.     

On sums of powers of integers

K. Thanigasalam has shown that for any positive integer k the sequence of positive integers represented by $\text{x}{}_2{}^2\text{+ x}{}_3{}^3\text{+ x}{}_5{}^5\text{+ x}{}_1{}^{\mathrm{k}}$ has positive density. Here we prove that the asymptotic density of this sequence is 1, a similar result is proved for the sequence represented by $\text{x}{}_2{}^2\text{+ x}{}_3{}^3\text{+ x}{}_5{}^6\text{+ x}{}_1{}^{\mathrm{k}}$.     

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.     

Formulas for sums of powers of integers by functional equations

This paper presents a direct and simple approach to obtaining the formulas forS _k(n)= 1 ^k + 2 ^k + ... +n ^k wheren andk are nonnegative integers. A functional equation is written based on the functional properties ofS _k (n) and several methods of solution are presented. These lead to several recurrence relations for the functions and a simple one-step differential-recurrence relation from which the polynomials can easily be computed successively. Arbitrary constants which arise are (almost) the Bernoulli numbers when evaluated and identities for these modified Bernoulli numbers are obtained. The functional equation for the formulas leads to another functional equation for the generating function for these formulas and this is used to obtain the generating functions for theS _k 's and for the modified Bernoulli numbers. This leads to an explicit representation, not a recurrence relation, for the modified Bernoulli numbers which then yields an explicit formula for eachS _k not depending on the earlier ones. This functional equation approach has been and can be applied to more general types of arithmetic sequences and many other types of combinatorial functions, sequences, and problems.     

Expressing powers of integers as the difference of squares

Three results are presented for expressing any power (greater than or equal to three) of any whole number as the difference of the squares of two other whole numbers.     

Representations of positive integers

The problem of representing integers as sums of terms of certain type is actual in number theory and its applications. We are interested in the average length of these expansions and the required number of auxiliary calculations. The paper deals with DBNS, chains, and the polyadic (factorial) expansions of positive integers.     

Partitions and sums of integers with repetition

A partition of N is called “admissible” provided some cell has arbitrarily long arithmetic progressions of even integers in a fixed increment. The principal result is that the statement “Whenever {A_i}_{i < r} is an admissible partition of N, there are some i < r and some sequence 〈x_n〉_{n < ω} of distinct members of N such that x_n + x_m?A_i whenever {m, n} ? ω″ is true when r = 2 and false when r ? 3.     

The Frobenius problem, sums of powers of integers, and recurrences for the Bernoulli numbers

In the Frobenius problem with two variables, one is given two positive integers a and b that are relative prime, and is concerned with the set of positive numbers NR that have no representation by the linear form ax+by in nonnegative integers x and y. We give a complete characterization of the set NR, and use it to establish a relation between the power sums over its elements and the power sums over the natural numbers. This relation is used to derive new recurrences for the Bernoulli numbers.     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.     

On sums of powers

Proceedings - Mathematical Sciences -     

Representing integers as sums or differences of general power products

We extend a result of Hajdu and Tijdeman concerning the smallest number which cannot be obtained as a sum of less than k power products of fixed primes. For this, we also generalize a classical result of Tijdeman concerning the gaps between integers of the form $p_{1}^{\alpha_{1}}\cdots p_{t}^{\alpha_{t}}$ where the p _i are primes, to the case where the numbers p _i are not necessarily primes.     

Representations of integers by linear forms in nonnegative integers

Let Ω be the set of positive integers that are omitted values of the form f = Σ_i=1ⁿa_ix_i, where the a_i are fixed and relatively prime natural numbers and the x_i are variable nonnegative integers. Set ω = #Ω and κ = max Ω + 1 (the conductor). Properties of ω and κ are studied, such as an estimate for ω (similar to one found by Brauer) and the inequality 2ω ≥ κ. The so-called Gorenstein condition is shown to be equivalent to 2ω = κ.