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Begin by writing the generating function for unrestricted partitions as follows: (1+x+x^2+x^3+…)(1+x^2+x^4+x^6+...)(1+x^3+x^6+x^9)⋯= 1+p(1)x+p(2)x^2+... Now change some of the coefficients from plus …

Lagrange's Theorem tells us that every integer can be written as the sum of at most four non-negative squares. Is it also true that, for example, every integer can be written as the sum of at most …

In his MathOverflow question "How thick is the reciprocal of the squares?" Kevin O'Bryant asks if a certain set, the reciprocal of the set of squares (identifying sets with power series in $F_2[[x]] …

First, some definitions: Let $A$ be a shorthand for an infinite sequence $( a(1), a(2), a(3), \dots )$ of positive integers, likewise $B$ is an infinite sequence of positive integers $( b(1), b(2), b …

Let $f(n)=1+x^n+x^{2n}$ Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers. Let $a(n)$ be the sequence of integers such that the coefficients of the series $f(a(1)) f …

In a chapter of Computational Algebra and Number Theory called Continued Fractions of Algebraic Numbers (Available at http://citeseerx.ist.psu.edu/viewdoc/download?rep=rep1&type=pdf&doi=10.1.1.135.107 …

A set of integers is called a basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of the set.(Nathanson, Additive Number Theory, page 7) …

A previous MO question asked for information about a number $x$ such that $\lfloor x^n\rfloor$ has the same parity as $n$ for all positive $n$. Answers to this post included two values of x which mee …

Start with the product $$(1+x+x^2) (1+x^2)(1+x^3)(1+x^4)\cdots$$ (The first polynomial is a trinomial..The others are binomials..) Is it possible by changing some of the signs to get a series all of …

This question is a variant of the question posed by Brian Hopkins. Let $\operatorname{pdist}(n)$ be the number of partitions of $n$ into distinct parts. Also, let $\operatorname{pdist}(S, n)$ denote …

Let $a(n)$ be the number of solutions of the equation $a^2+b^2\equiv -1 \pmod {p_n}$, where $p_n$ is the n-th prime and $0\le a \le b \le \frac{p_n-1}2$. Is the sequence $a(1),a(2),a(3),\dots$ non-dec …

On page 384 of the book "Number Theory:Volume 1:tools and Diophantine Equations" by Henri Cohen there is reference to the fact that: "It has been proved by Schinzel, Mordell nd successors that such an …

Let $F_n$ denote the $n$th Fibonacci number. Then $\prod\limits_{i=1}^{\infty}(1-x^{F_i})$ is a series all of whose coefficients are either $-1$, $0$ or $+1$. The sequence of the coefficients in thi …

Let p(n) be the number of unrestricted partitions of n. p(0) is taken to be 1. Let set 1 and set 2 be two empty sets. Here's an algorithm. Put p(n) into set 1. On each successive step, k=1,2,3,..., …

Let $f(n)=1+x^n+x^{2n}+...+x^{n^2}.$ Let $p(x)$ be $1+x+x^2+x^5+x^7+...$ where the exponents are the pentagonal numbers. Let $a(n)$ be the sequence of integers such that the coefficients of the seri …