An elementary proof is given of the Andrews-Krammer-Crandall formula for the number r_3(n) of representations of a positive integer n as the sum of three squares.

We give an explicit formula for the value of the Bernoulli polynomial B2k(t) when t is a rational number in the interval (0, 1). When t = 12, 31, 32, 41, 34, 61, 65 the value of B2k(t) is known explicitly. In 1938 Emma Lehmer asked for the value of B2k(t) when the denominator of t is 5, 8, 10, or 12. We apply our formula to determine B2k(t) when the denominator of t is 5, 8, 10, and 12 thereby answering Lehmer's 87 year old question. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This article shows how two classical results about polynomials with trigonometric roots may be used together to create interesting new formulas for trigonometric sums. Readers are encouraged to find further new trigonometric formulas by applying the ideas in this article to other polynomials having trigonometric roots.

Chan and Cooper proved that if the integers c(n) (n=0,1,2,… ) are given by ∑ _n=0^∞ c(n)q^n = ∏ _n=1^∞1/( 1-q^n) ^2( 1-q^3n) ^2, then ∑ _n=0^∞ c(2n+1)q^n = 2 ∏ _n=1^∞( 1-q^2n) ^4( 1-q^6n) ^4/( 1-q^n) ^6( 1-q^3n) ^6. We prove many other results of this type and apply them to the determination of congruence properties of the coefficients.

Abstract We present a short arithmetic proof of the theorem of Chan, Long, and Yang proved in the Monthly in 2011, which gives explicit formulas for integers x and y such that , where p is a prime satisfying .

For integers a and b (not both 0) we define the integers c(a,b;n) (n=0,1,2,… ) by ∑ _n=0^∞ c(a,b;n)q^n = ∏ _n=1^∞( 1-q^n) ^a (1-q^2n)^b (|q|<1). These integers include the numbers t_k(n) = c(-k,2k;n) , which count the number of representations of n as a sum of k triangular numbers, and the numbers (-1)^n r_k(n) = c(2k,-k;n) , where r_k(n) counts the number of representations of n as a sum of k squares. A computer search was carried out for integers a and b , satisfying -24≤ a,b≤ 24 , such that at least one of the sums 0.1 ∑ _n=0^∞ c(a,b;3n+j)q^n, j=0,1,2, is either zero or can be expressed as a nonzero constant multiple of the product of a power of q and a single infinite product of factors involving powers of 1-q^rn with r∈{1,2,3,4,6,8,12,24} for all powers of q up to q^1000 . A total of 84 such candidate identities involving 56 pairs of integers ( a , b ) all satisfying a≡ b (mod 3) were found and proved in a uniform manner. The proof of these identities is extended to establish general formulas for the sums (0.1). These formulas are used to determine formulas for the sums ∑ _n=0^∞ t_k(3n+j)q^n, ∑ _n=0^∞ r_k(3n+j)q^n, j=0,1,2.

In the 18th century, the problem of finding the sum of the reciprocals of the squares was known as the Basel problem. It was first solved by Euler, who, in 1735, showed the sum evaluated to π2/6. We go beyond the Basel problem by evaluating the sum of 1/n2 where n≡±b (moda), for some integers b and a. In particular, we examine the special cases n≡±1 (mod5) and n≡±3 (mod16).

Abstract In this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant $-31$ and Fourier coefficients of a weight $16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant $-23$ .

Let Fm denote the set of positive-definite primitive integral quadratic forms in m variables. Let f,g∈Fm. In this paper we introduce a new concept, namely that of g being derivable from f. This concept is based on a certain theta function identity being valid. A consequence of this concept is that if g is derivable from f then the representation number of g can be given in terms of that of f. Many examples are given, especially for diagonal ternary quadratic forms.

SummaryA prime dividing a composite Fermat number is called a Fermat prime divisor. Such a prime p must be congruent to 1 modulo 4, and so, by the Fermat–Girard theorem, there exists integers R and S such that p = R2 + S2. We derive a necessary and sufficient condition for p to be a Fermat prime divisor in terms of the integers R and S.

The purpose of this paper is to present some examples of positive-definite integral nondiagonal quaternary quadratic forms whose representation numbers can be determined explicitly using the theory of modular forms. Very few such examples appear in the literature. The seven forms presented were selected because they each belong to a genus containing exactly two form classes for which the single genus mate is a diagonal form whose representation number has been determined recently.

A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$ there are integers $x_{1},\ldots ,x_{n}$ such that $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}=p$. We determine all possible prime-universal ternary quadratic forms $ax^{2}+by^{2}+cz^{2}$ and all possible prime-universal quaternary quadratic forms $ax^{2}+by^{2}+cz^{2}+dw^{2}$. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature.

Summary.We show how Liouville’s formulas for the number of representations of a positive integer by the forms x12+x22+2x32+2x42,x12+2x22+2x32+4x42,x12+x22+x32+4x42,x12+x22+4x32+4x42, and x12+4x22+4x32+4x42 follow in a simple systematic way from a beautiful identity of Jacobi using some elementary relationships between the infinite series P(x)=1+2x+2x4+2x9+⋯ and Q(x)=1−2x+2x4−2x9+⋯ given by Gauss.

In 1997 Jagy, Kaplansky and Schiemann determined that there are at most 913 (classes of) primitive, positive-definite, integral ternary quadratic forms ax2 + by2 + cz2 + dyz + ezx + fxy which are regular. In this paper the positive integers represented by these 913 ternary forms are given.

Summary The Pythagorean theorem gives a fundamental relationship between the lengths of the three sides of a right-angled triangle. We make the case for which theorems of elementary Euclidean geometry should be called the “Pythagorean theorem for an arbitrary triangle” and the “Pythagorean theorem for a quadrilateral.” In making this presentation, we stress the use of the basic properties of triangles over use of the cosine formula.

Lagrange's theorem tells us that the quadratic form represents all positive integers. What about the more general form, where a, b, c, and d are positive integers? We provide a gentle introduction to this question by posing and answering some natural questions about the possible integers that can represent.