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Google turns up this "Mock Putnam Exam" from the U[niversity] of I[llinois], whose unattributed second problem asks to show that $[(\sqrt2+1)^n]$ has parity opposite to that of $n$ for each $n=1,2,3,\ …

Yemon Choi (henceforth "YC") answered the question to within a constant factor with a function $f_\delta$ showing that $\alpha(E_\delta) \leq 2/\delta$ where $E_\delta$ is an interval of measure $1-\d …

Gauss's hypergeometric formula for the AGM can also be interpreted in terms of a complete elliptic integral $\int_0^{\pi/2} \phantom. d\theta / \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta}$. There's …

The ATLAS of Conway et al. has this information for $S_n$ for each $n = 5, 6, 7, \ldots, 13$. For $S_7$, look up $A_7$ in the main table to find the characters of Aut$(A_7)=S_7$, then check the appen …

This can't be right in general. For example $$ x = 0.101000001000000000000000001\ldots = \sum_{k=0}^\infty 10^{-3^k} $$ has plenty of lower approximants with exponent just under 3 (the partial sums) …

Yes, there are uncountably many examples (maybe I should have mentioned this variation with my previous answer). For example, $\sum_{k=0}^\infty b_k 10^{-3^k}$ with each $b_k=1$ or $2$. There are un …

The reference Weinberger, P. J.: Exponents of the class groups of complex quadratic fields, Acta Arith. 22 (1973), 117–124. was what Matthias Schütt and I cited for the fact that "there is at most …

This is addressed in various texts, including the Silverman book cited by A.Lozano-Robledo and (if memory serves) Chapter VII of Serre's A Course in Arithmetic. Explicitly, let $E_4$ and $E_6$ be the …

The table in that Mathworld page suggests that $p(d) \rightarrow 0$ as $d \rightarrow \infty$. That page also gives a formula for $p(d)$ in terms of a definite integral: $$ p(d) = 1 - \left[ \int_0^\i …

We use complex numbers to prove that there are at most $4$ such points unless both transmitter and receiver are at the center. Identify the circular room with the unit circle $|z|=1$ in the complex p …

For any distinct integers $x_1,\ldots,x_m$ you can take the matrix whose $i$-th row $(1 \leq i \leq m)$ is $(1,x_i,x_i^2,\ldots,x_i^{n-1})$. Each $n \times n$ submatrix is Vandermonde with distinct ro …

Presumably the indices $i$ in $A_i$ are taken mod $n$, so "$A_{n+1}$" is to be identified with $A_1$. This must be a known isoperimetric inequality, but it's easier to prove than to find in the litera …

Trace is additive, and ${\rm Tr}(u)={\rm Tr}(u^2)$ for all $u$, so $ax^2+bx$ has the same trace as $(a+b^2)x^2$. Therefore the sum is $2^r$ if $a=b^2$ and zero otherwise. In general, for a polynomial …

A couple of years after my Ph.D. thesis (whose main result is the infinitude of singular primes, i.e. $P(x) \rightarrow \infty$ as $x \rightarrow \infty$), Kaneko published a result[1] that let me obt …

[edited to include more examples of the construction and fix a typo] This can also be seen purely in terms of lattices or quadratic forms. As you note, for any $n$ the vectors in the slice $\sum_{i=0 …