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The original and some subsequent proofs of the irrationality of $\pi$ implicitly use the following lemma: If there is a sequence $P_n(x) \in \mathbb{Z}[x]$ such that $P_n(\alpha)>0$ and $P_n(\alpha)=o …

I was thinking about how the standard proofs that $\pi$ is irrational use variations on $sin(\pi) =0$, in contrast to Apéry's proofs that $\zeta(2),\zeta(3)$ are irrational (the first of course also p …

Short expansion of my comment on what's immediately achievable when assuming the $abc$-conjecture. Proporistion. Assuming the $abc$-conjecture, we have $$v_p((p-1)!+1)=o(p).$$ Proof. One formula …

Given a polynomial $f$, it is known believed that the number of smooth values of $f$ has a positive proportion (for fixed $u$, $\lim_{X\rightarrow\infty} \frac{|\{ n < X\ :\ f(n)\ is\ X^u\ smooth \}|} …

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and w …

So, how does one construct a galois representation from a Maass form? For a modular cusp eigenform, I am familiar with the work of Eichler-Shimura, Deligne, Deligne-Serre, and realize these are diffe …

$\newcommand\Q{\mathbf{Q}}$ $\newcommand\Qbar{\overline{\Q}}$ $\newcommand\Gal{\mathrm{Gal}}$ $\newcommand\C{\mathbf{C}}$ $\newcommand\Sym{\mathrm{Sym}}$ $\newcommand\E{\mathcal{E}}$ $\newcommand\Bett …

Of course not. But after reading a bit, some points make me believe it should be: Let $S$ be a nice$^{\*}$ surface defined over $Spec\ \mathbb{Z}$. The Brauer group $Br(S\otimes \bar{\mathbb{Q}})$ …

The case of even powers was (asked and) solved by P. T. Bateman in "Problem E 2051", Amer. Math. Monthly 76 (1969), the solution being that the powers 2, 4, 8 are the only even powers of $\Theta$ that …

Given a galois extension of number fields $L/K$ of even degree, set $n_0=\text{lcm} (\{[L_v:K_v] : v \in M_K \})$ ($L_v$ is any completion corresponding to a place dividing $v$). Does $2$ divide $n_0 …

[big edit] (1) Let $L/K$ be as above. Take any non-identity element $\sigma \in G_{L/K}$, and let $F=L^{<\sigma>}$ be the fixed field of the cyclic subgroup generated by $\sigma$. By your comment abo …

Tunnel's result on the congruent number problem hinges on the fact that there are modular forms with fourier coefficients related to the values $L(E_n,1)$. Is there an interesting function that has c …

I'll begin by saying that I'm not sure what I want to ask specifically, but pretty sure what in general, so please don't hold my misunderstandings against me, but do comment on them. I know that the …

Let $d$ be a positive integer greater than 2. Define an equivalence relation on monic integer polynomials of degree $d$: $f\sim g \iff f(k_1 x+k_2)=g(k_3 x+k_4)$ for some integers $k_1,...,k_4$. I …

It is possible to shorten Bertrand's Postulate's proof so it proves only the above. We can throw away the usually-proven upper bound on the primoral. Explicitly, following Wikipedia's "Proof of Bertra …