20 problems

Let $(a,b)$ be coprime, let $\mathcal{U}_{a,b}$ denote the expansion of $(\mathbb{N},+,0,1)$ by the associated Ulam predicate, and let $\operatorname{Th}(\mathcal{U}_{a,b})$ be its…

Let $(a,b)$ be coprime, and let $\mathcal{U}_{a,b}$ be the expansion of $(\mathbb{N},+,0,1)$ by the predicate for the associated Ulam sequence. Ulam dichotomy. For every coprime…

Let $(a,b)$ be coprime, let $U(a,b)$ be the associated Ulam sequence, and write its consecutive gaps as $g_k$. Condition (R3) means that there exists $B$ such that all gaps $g_k$ a…

Let $(a,b)$ be coprime, and let $\mathcal{U}_{a,b}$ denote the expansion of $(\mathbb{N},+,0,1)$ by the predicate for the associated Ulam sequence. Non-interpretability of multipli…

For any integer $n$, let $\operatorname{eval}_n:\mathbb{Z}[X]\to\mathbb{Z}$ be the evaluation homomorphism defined by $X\mapsto n$. Let $U(1,X)$ be the polynomial-ring analogue of…

Let $U(1,X)$ denote the polynomial-ring analogue of the Ulam sequence, and let $a_i,b_i,c_i,d_i$ be its coefficients. For real $X$, reduction modulo $3X+\lambda'$ is taken using re…

For each $n\geq4$, write the Ulam sequence $U(1,n)$ as a union of integer intervals, with the coefficients below required to be integer coefficients. Block decomposition conjecture…

For integers $1\leq a<b$, let $U(a,b)$ be the Ulam sequence whose later terms are the smallest integers representable as sums of two distinct prior terms in exactly one way. For a…

Let $U(1,2)$ be the Ulam sequence, and for a positive real number $\lambda$ interpret reduction modulo $\lambda$ by taking representatives in an interval of length $\lambda$. Gibbs…

Given integers $1\leq a<b$, the Ulam sequence $U(a,b)$ starts with $a,b$, and each subsequent term is the smallest integer expressible as the sum of two distinct prior terms in exa…

Let $a,b$ be parameters of a two-generator Ulam sequence $\mathcal{U}(a,b)$. Say that two such sequences follow the same pattern when their corresponding structural behavior agrees…

Let $a,b$ be relatively prime positive integers, with one of $a,b$ even. For an Ulam sequence $\mathcal{U}(a,b)$, Finch's even-term conjecture. If $a$ is even, then…

Let $a,b$ be relatively prime, with $a$ even and $b>2a$ odd. The modified Ulam sequence $\mathcal{V}(a,b)$ has explicitly listed even terms in the cases $a\in\{4,6,8,10,12,14,16\}$…

Let $a<b$ be relatively prime positive integers. Let $\mathcal{U}(a,b)$ denote the Ulam sequence generated by $a,b$, let $\mathcal{V}(a,b)$ denote the corresponding modified Ulam s…

Let $n\geqslant 5$ be odd, and let $\mathcal{V}(2,n)$ be the greedy sequence whose initial terms are $2,n$ and whose subsequent terms are the least larger integers having a unique…

Finite-even-terms conjecture. The set $U(a,b)$ contains finitely many even terms if and only if one of the following holds:

Numerical rigidity conjecture. There exists an integer $N_0>1$ such that, for every $n\geq N_0$, the endpoints are given by

Rigidity Conjecture. For every positive integer $a$, there exists an integer $L\geq1$ such that, for every $c\in(\mathbb{Z}/L\mathbb{Z})^\times$, there exists a positive integer…

Interval decomposition conjecture. There exist integer coefficients $m_i,p_i,k_i,r_i$ such that, for every integer $n\geq 4$,

Let $(a_1,a_2)=(2,n)$ with $n\geq 5$ odd, and construct the associated Ulam-type sequence by the usual rule: each subsequent term is the least integer that is the sum of two distin…