Related papers: Mixed sums of triangular numbers and certain binar…

Commutative complex numbers of the form u=x+\alpha y+\beta z+\gamma t in 4 dimensions are studied, the variables x, y, z and t being real numbers. Four distinct types of multiplication rules for the complex bases \alpha, \beta and \gamma…

In this work, we define a triangle area number to be the area number of a triangle whose sides have integer lengths, and whose area is a rational number. In Result 3, on page 17, we prove that every triangle area number is in fact an…

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$…

It is well-known that Lagrange's four-square theorem, stating that every natural number may be written as the sum of four squares, may be proved using methods from the classical theory of modular forms and theta functions. We revisit this…

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a…

In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer…

Let $\sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$…

The question of which triangular numbers have a decimal representation containing a single repeated digit seamed to be settled since at least the 1970s: Ballew and Weger provided a complete list and a proof that these are the only numbers…

Magic squares are well-known arrangements of integers with common row, column, and diagonal sums. Various other magic shapes have been proposed, but triangles have been somewhat overlooked. We introduce certain triangular arrangements of…

In this methodological paper, we first review the classic cubic Diophantine equation $a^3 + b^3 + c^3 = d^3$, and consider the specific class of solutions $q_1^3 + q_2^3 + q_3^3 = q_4^3$ with each $q_i$ being a binary quadratic form. Next…

In this article we present set of infinite natural numbers which satisfies the conjecture $3n+1$.

We give some new canonical representations for forms over $\cc$. For example, a general binary quartic form can be written as the square of a quadratic form plus the fourth power of a linear form. A general cubic form in $(x_1,...,x_n)$ can…

In this paper we study mixed sums of primes and linear recurrences. We show that if m=2(mod 4) and m+1 is a prime then $(m^{2^n-1}-1)/(m-1)\not=m^n+p^a$ for any n=3,4,... and prime power p^a. We also prove that if a>1 is an integer, u_0=0,…

For a natural number $m$, generalized $m$-gonal numbers are defined by the formula $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper, we determine a criterion on $a,b,c,m$ for which the weighted ternary sum…

Let $\Bbb Z$ and $\Bbb Z^+$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb Z^+$ let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x+1)/2+by(y+1)/2+cz(z+1)/2+dw(w+1)/2$…

An important unsolved problem in Diophantine number theory is to establish a general method to effectively find all solutions to any given $S$-unit equation with at least four terms. Although there are many works contributing to this…

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 25 is the sum of at most three natural numbers whose…

We say a positive integer is a sum of three nonunit squares if it is a sum of three squares of integers other than one. In this article, we find all integers which are sums of three nonunit squares assuming that the Generalized Riemann…

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$…