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Let $f(x)$ be a nonconstant polynomial over $\mathbb Z$. Must $f$ have a zero in $\mathbb F_p$ for some prime number $p$? More generally, let $f_1,\dots,f_k$ be such polynomials, must there exist a …

It is actually an old conjecture of Erdős, Mollin, and Walsh that the pattern you have noticed does indeed go on forever, i.e., there are no three consecutive powerful numbers.

Let us call a set $D\subseteq\mathbb Z$ residue-class dense if for each residue class $[a]_n=\{kn+a\mid k\in\mathbb Z\}$, there is a residue class $[b]_m$ with $[b]_m\subseteq [a]_n\cap D$. Using the …

For each $k\ge 1$ there is a sequence $x_{1,k},\ldots,x_{k,k}$ of positive integers such that for all nonnegative integers $ a_i $, $$ \sum_{i=1}^k a_i x_{i,k} =2\sum_{i=1}^k …

I can deduce some results about this from the prime number theorem or from results about the primorial function $p\#$ but I'm wondering what the state of the art is: Given integers $0\le a_1<a_2<\dots …

The condition $\lambda>1$ is sufficient and, at least almost, necessary: To clarify, the space of irrational numbers $(0,1)-\mathbb Q$ is homeomorphic to $\omega^\omega$ under the map that sends $\fr …

If $q\in x+C$, i.e., $q=x+c$ for some $c\in C$, then $x=q-c$, i.e., $x\in q-C$. Now $q-C$ is a closed set of measure zero and moreover it is a $\Pi^0_1$ class if $q\in\mathbb Q$. Thus each weakly 1-ra …

Consider the equation $$6x+3y+2z=13$$ for $x$, $y$, $z$ nonnegative integers, with the constraints $$x=0\implies y=0,$$ $$x=0\implies z=0.$$ The set of solutions $(x,y,z)$ is a kind of quasivariety wh …

Euler in 1769 conjectured that for all integers n and k greater than 1, if the sum of k nth powers of positive integers is itself a nth power, then k is greater than or equal to n: $$a^n_1 + a^n_2 …

Let $c$ be a constant such that $$\mathrm{PA}\not\vdash K(x)\ge c$$ for all binary strings $x$, where $K$ is Kolmogorov complexity. Such a $c$ exists by Chaitin's Incompleteness Theorem and the linked …

For many variants of this question the answer seems to be not known but at least this question in the comments More generally, are there infinitely many primes such that at least one non-trivial …

Chebyshev's bias says that there are slightly more non-Pythagorean primes than Pythagorean primes (although the limiting frequency is the same).

Let $C_k$ be the set of partitions of $n$ containing $k$. Following @MaxAlekseyev's point we have, $$P(n-1)+P(n-2)-P(n)=|C_1|+|C_2|-P(n),$$ $$=|C_1\cap C_2|+|C_1\cup C_2|-P(n)$$ This is the # of part …

As is known, the set $\{1,\ldots,n\}$ has $2^n$ many subsets and $B_n$ (the $n$th Bell number) many partitions, where clearly $B_n<2^{2^n}$ and it is actually known that $B_n<n^n$ for large $n$. A n …

As I understand your question, we have random variables $X_n$, $n\in\mathbb N$ taking values in $\{0,1\}$. We intuitively think of $X_n=1$ as ``$n$ is prime'', but other than that this has little to …