MathOverflow

Is it possible to use the Schur complement determinant calculation procedure to prove the determinant of transpose identity? The base case would be simple with dimensions n = 1, where det(a) = det(a_T). I am only a high school student, so I may not be...

In his 2004 paper "Prove or Disprove: 100 Conjectures from the OEIS"(arXiv:math/0409509), Ralf Stephanconjectured (Conjecture 19):
If $\mathrm{PCF}(n) = \mathrm{PCF}(n+1) \equiv 1 \pmod{2}$, then$n \equiv 1 \pmod{24}$.
Here $\mathrm{PCF}(n)$ denotes the period length of the continued fractionexpansion of $\sqrt{n}$, for positive non-square $n$....

Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view was my motivation for asking this question almost...

Below is an apparently very basic result in additive combinatorics:
Let $A,B \subseteq \mathbb{Z}$ be finite sets. Let $A+B = \{a+b : a \in A, b \in B\}$ be their sum-set. Then $|A+B| \geq |A| + |B| - 1$, with equality if and only if $A,B$ are arithmetic progressions with the same common difference....