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This is essentially a reference request/name inquiry. Is there a name for the set $M_k$ formed by $k$ by by $k$ matrices with non-negative integer entries and positive values on the diagonal? Related, …

Another major result from this year was Rachel Greenfeld and Terry Tao's disproof of the periodic tiling conjecture. The conjecture essentially said that if one has a single tile which aperiodically t …

One from just a few days ago is Justin Gilmer's breakthrough on the union-closed conjecture, also known as Frankel's conjecture, which says that if one has a finite family of sets which are closed und …

Andrew Granville and Jennifer Granville's "Prime Suspects" is a comic book which despite being phrased as a murder mystery does a surprisingly lucid job connecting ideas involving primes with ideas in …

Sieve theory as such is generally considered to have started with Brun's 1915 and 1919 papers. The titles are "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare" and ""La série $1/5+1/7+1/ …

There's been a lot of work on unconditional results of this sort. Rosser and Schoenfeld showed in a 1962 paper that one can take $$\dfrac{e^{-\gamma}}{\log x} \left(1- \frac{1}{2\log^2 x} \right) < …

This should be provable by standard although laborious methods. What follows is a proof sketch (I have not checked all the computational details but this method should work). We recall a few basic fac …

Theorem: For any constant $c$ there are infinitely many primes $p_k$ such that there are at least $c$ primes between $p_k^2$ and $p_{k+1}^2$. Proof: Fix a $c$. Assume that for sufficiently large $k$ …

In general, very few prime factors in an odd perfect number can be of the form $n^2+1$. In particular, if N is an odd perfect number then $\frac{\sigma(N)}{N}=2$, and for any $m$ (perfect or not), $\f …

It is a standard conjecture that if one replaces the 3 in the Collatz function with some fixed $k>3$, then it will have sequences which go to infinity. And in fact, these seem to occur at very small v …