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Mar
26
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Nov
3
comment Sphere Covering Problem
@ThomasAhle: I never learned German before. I got a copy of Fejes Toth's book from the library of my university, and used google translate to get the main idea of the proof. If I remember correctly, the bound itself is an application of Jensen's inequality.
Sep
22
accepted Square-free value of $n^2+1$ without large prime factors
Sep
22
comment Square-free value of $n^2+1$ without large prime factors
Thank you so much, Prof. Elkies! I guess that the density should be $c_n(1-\log 2)$(where $c_n$ is the density of square free $n^2+1$) , but it might be very hard to prove such a result.
Sep
17
awarded  Favorite Question
Sep
13
asked Square-free value of $n^2+1$ without large prime factors
Sep
5
revised Some questions on the formation of the BSD conjecture
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Aug
14
awarded  Enlightened
Aug
14
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Mar
26
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Dec
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Nov
10
comment How to find certain quadratic curves over $\mathbb{Q}$
Thanks. It is likely that there are solutions other than the unit circle. It is possible to generate a family of such curves?
Nov
10
asked How to find certain quadratic curves over $\mathbb{Q}$
Nov
6
comment Hasse's theorem on elliptic curves over finite fields
I think multiplicity-one theorem forbids such a construction. Wiles proves that $\lambda_p+\bar{\lambda_p}$ comes from the coeffecient of some modular form of weight 2, and multiplicity-one theorem asserts that coefficients from two modular (eigen)forms satisfy $a_p^2=b_p^2$ for almost all prime $p$ if and only if $a(p)=b(p)\chi(p)$, where $\chi$ is some quadratic character. Your construct forces $\chi(p)=-1$ for almost all prime $p$, which is not possible.
Nov
2
comment Identities related to hypergeometric functions
I use Clausen's formula and linear transformation formula for the deduction, and things might go bad for the hypergeometric functions are multi-valued around its singularities. A numerical computation on Mathematica suggest that A is true for $0<k<\sqrt{2}-1$ while B is true for $k>\sqrt{2}-1$.
Nov
2
comment Identities related to hypergeometric functions
Hmm. A and B cannot be simultaneously true for the same k. I am wondering whether a transformation formula exists. RHS has a singular point at $\sqrt{2}-1$, and I need such a formula for analytic continuation, and I hope this would help.
Nov
2
revised Identities related to hypergeometric functions
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Nov
2
comment Identities related to hypergeometric functions
Eh, $K^{\prime}$ does not represent the derivative of $K$ in elliptic integral theory. It is $K(\sqrt{1-k^2})$
Nov
2
asked Identities related to hypergeometric functions
Oct
7
comment Some questions on the formation of the BSD conjecture
Hmm. I have read a nice paper of Franz Lemmermeyer with the title "Conics: A poor man's elliptic curves", where such analog is discussed. But I am still wondering how Tate found the analog of regulator on the elliptic curves. I don't think numerical computations will help to establish the refined BSD conjecture.