Related papers: On reaping number having countable cofinality
In the setting of constructive mathematics, we suggest and study a framework for decidability of properties, which allows for finer distinctions than just "decidable, semidecidable, or undecidable". We work in homotopy type theory and use…
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…
We show that for any nontrivial knot $K$ and any natural number $n$ there is a diagram $D$ of $K$ such that the unknotting number of $D$ is greater than or equal to $n$. It is well known that twice the unknotting number of $K$ is less than…
We obtain an upper bound on the number of extended irreducible q-ary Goppa codes of degree $r$ and length $q^n+1$, where $q=p^t$ and $n$ and $r>2$ are prime numbers.
A partition into distinct parts is refinable if one of its parts $a$ can be replaced by two different integers which do not belong to the partition and whose sum is $a$, and it is unrefinable otherwise. Clearly, the condition of being…
Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~…
We force over a model of AD to obtain the consistency of the Galvin number having countable cofinality.
Cantor's diagonal method is traditionally used to prove the uncountability of the set of all infinite binary sequences. This paper analyzes the expressive limits of this method. It is shown that under any constructive application --…
We revisit the problem of computing the spreading and covering numbers. We show a connection between some of the spreading numbers and the number of non-negative integer 2x2 matrices whose entries sum to d, and we construct an algorithm to…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…
Let $b \geq 3$ be a positive integer. A natural number is said to be a base-$b$ Zuckerman number if it is divisible by the product of its base-$b$ digits. Let $\mathcal{Z}_b(x)$ be the set of base-$b$ Zuckerman numbers that do not exceed…
In this paper we apply a geometric covering method to study the number of ends on shrinkers. On one hand, we prove that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree. On the other…
We study Hoeffding decomposable exchangeable sequences with values in a finite set D. We provide a new combinatorial characterization of Hoeffding decomposability and use this result to show that, if the cardinality of D is strictly greater…
We consider the problem of characterizing solutions in $(x, y)$ to the equation ${x \choose y}={{x-a} \choose {y+b}}$ in terms of $a$ and $b$. We obtain one simple result which allows the determination of a ratio in terms of $a$ and $b$…
Consider an irreducible bilinear form $f(x_1,x_2;y_1,y_2)$ with integer coefficients. We derive an upper bound for the number of integer points $(\mathbf{x},\mathbf{y})\in\mathbb{P}^1\times\mathbb{P}^1$ inside a box satisfying the equation…
We determine the sets definable in expansions of the ordered real additive group by generalized Cantor sets. Given a natural number $r\geq 3$, we say a set $C$ is a generalized Cantor set in base $r$ if there is a non-empty…
A common question from students on the usual diagonalization proof for the uncountability of the set of real numbers is: when a representation of real numbers, such as the decimal expansions of real numbers, allows us to use the…
Assuming the existence of a strong cardinal $\kappa$, a weakly compact cardinal $\lambda$ above it and $\gamma > \lambda,$ we force a generic extension in which $\kappa$ is a singular strong limit cardinal of any given cofinality $\delta$,…
It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate…
We prove that assuming suitable cardinal arithmetic, if B is a Boolean algebra every homomorphic image of which is isomorphic to a factor, then B has locally small density. We also prove that for an (infinite) Boolean algebra B, the number…