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### A function asymptotical equivalent with the prime counting function?

This is closely connected to Schinzel's Hypothesis H and the Bateman-Horn conjecture. A slightly different way to phrase this question would be to consider the two polynomials $(x, x^2 + 4)$, and look ...
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### Tate gamma factor as a principal value integral

For the second question, let $\varpi$ be a uniformizer, and set $\mathfrak p = \varpi \mathcal O_F$. Let's define the conductor of $\chi$ to be $f$ if $\chi$ is trivial on $1+\mathfrak p^f$, but not ...
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### Change of fractional numbers from one base to another

I'll assume you know how to write a fraction $A/B$ in decimal form, i.e. $A/B=(0.d_1d_2...)_{10}$. Then, given that $x=(0.a_1a_2...)_b=\sum_{i=1}^\infty a_i/b^i,$ just write each successive term in ...
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### How to find all natural numbers $n$ such that $a \mid n$ implies $(a+1) \mid (n+1)$?

Let $n$ be a good number. Claim 1. $n$ is not a composite number. Claim 2. $1$ is a good number (which is obvious since $(1+1) \mid 2$). Claim 3. All primes but $2$ are good. In conclusion, we've ...
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Accepted

Yes: $\pi_1^{\mathrm{et}}(\mathrm{Spec}(\mathbb{Z}_{(p)})$ is $\mathrm{Gal}(K/\mathbb{Q})$ where $K$ is the maximal extension of $\mathbb{Q}$ unramified at $p$. This is not special to $\mathbb{Z}_{(p)}... • 48.1k 3 votes Accepted ### Field over which CM endomorphism is defined Yes. Let$E/K:y^2=x^3+ax+b$such that$End(E)$is isomorphic to an order$O$in$\Bbb{Q}(\sqrt{-D})$. For any$\alpha\in O$let$\phi\in End(E)$be the "multiplication by$\alpha$" ... • 67.7k 2 votes ### Show that there does not exist integers$n_{1}, \ldots, n_{8}$, not necessarily distinct, such that$n_{1}^{4}+\cdots+n_{8}^{4}=1993$. HINT.-You have$n^4\equiv0\pmod{16}$when$n$is even and$n^4\equiv1\pmod{16}$when$n$is odd then taking modulo$16$you have$ n_{1}^{4}+\cdots+n_{8}^{4}\le8$. But$1993\equiv 9\pmod{16}$• 25.3k 4 votes Accepted ### Show that there does not exist integers$n_{1}, \ldots, n_{8}$, not necessarily distinct, such that$n_{1}^{4}+\cdots+n_{8}^{4}=1993$. You have the right idea, but using modulo$16$instead works better (as also suggested in Arturo Magidin's comment). Similar to modulo$8$, in modulo$16$, every even number raised to the power$4$is ... • 39.4k 0 votes ### Units in ring of integers of$5^{th}$cyclotomic field Consider $$f: O_K^\times\to \Bbb{Z}[\varphi]^\times, z\to |z|^2$$ where$\varphi= \frac{1+\sqrt5}{2}$.$\ker f = \{ a\in O_K^\times,\forall \sigma\in Hom_\Bbb{Q}(K,\Bbb{C}),|\sigma(a)|=1\} = \mu_{10}$... • 67.7k 1 vote ### The set of integers$n$expressible as$n=x^2+xy+y^2$Denote$a^2+ab+b^2$by$L(a, b)$. Numbers of the form$L$are known as Loeschian numbers. What you are trying to prove is that the set$S$of Loeschian numbers is multiplicative. Here's a direct proof ... • 2,388 1 vote ### Proving that this infinite product is convergent By partial summation, one can deduce the existence of a constant$C_0$such that $$\sum_{p\le z}{w(p)\over p}=K\log\log z+C_0+O\left(1\over\log z\right)$$ As$t\to0^+$, we know$-\log(1-t)=t+O(t^2)$,... • 3,360 3 votes ### Prove that$\frac{2022}{n} + 4n$is a perfect square iff$\frac{2022}{n} - 8n$is a perfect square You can greatly reduce the number of cases to check by working modulo$4$. We have $$2022 = 2\cdot 3 \cdot 337 \equiv 2\cdot 3\cdot 1 \pmod 4.$$ Since a square must be$0$or$1$mod$4$, we can ... • 5,721 0 votes ### Looking for number systems with solutions for$x^2 = 0$other than$x = 0$Let$n\geq2$be an integer, and consider the ring$\mathbb{Z}_{n^2}$(the integers modulo$n^2$). Then clearly$n<n^2$, so$n$is not the zero element in$\mathbb{Z}_{n^2}$, but $$n^2\equiv0\pmod{n^... • 2,467 1 vote ### Looking for number systems with solutions for x^2 = 0 other than x = 0 Consider the commutative ring \Bbb Z/25\Bbb Z. Then x^2=0 has five solutions, namely$$ x=0,5,10,15,20.$$The ring$\Bbb Z/n\Bbb Z$is a field if and only if$n$is prime. The equation$x^2=0\$ ...
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