14,826 reputation
14484
bio website math.brown.edu/~jhs
location Brown University Mathematics Department
age
visits member for 4 years, 6 months
seen 4 hours ago
Professor of Mathematics, Brown University. Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.

16h
comment Is the Jaccard distance a distance?
Would it be possible (and/or appropriate) to put a link on the Wikipedia page to this MO question and answer? I'm hoping someone here knows more about Wikipedia guidelines than I do regarding such links.
2d
comment N-th root of unity in N-th division field of abelian variety?
@ACL Very nice!
Jun
23
comment How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$ if it's not open problem?
@RobertIsrael Very tricky. But it's not absolutely convergent, right, because $\tan(\pi n!/e) = (-1)^{n+1}/(n+1) + O(1/n^2) \pmod{\mathbb Z}$?
Jun
23
comment How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$ if it's not open problem?
Actually, it's not clear that there's even a polynomial bound, i.e., $\sum_{n\le X}\tan(n!)=O(X^K)$ for some $K>0$. Rigorously, probably the best one can do is use transcendence estimates for $\pi$ to bound $n!$ away from multiples of $\pi/2$ where $\tan$ blows up, which is going to yield an effective, but horrible, upper bound in terms of $X$.
Jun
23
comment How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$ if it's not open problem?
It's probably more interesting to look at the values of something like $\sup_{n\le N} \frac{\log|f(n)|}{\log n}$ and see if it looks as if there might be a polynomial bound for $|f(n)|$.
Jun
23
comment How do I evaluate this sum :$\sum_{n=0}^{\infty} \frac{\sin(n!)}{\cos(n!)}$ if it's not open problem?
Presumably the set of values $\{ n! \bmod 2\pi\mathbb Z\}$ is dense in the interval $[0,2\pi]$, in which case $\tan(n!)$ will be arbitrarily large, infinitely often. So your series is highly unlikely to be absolutely convergent. (Whether one can prove this or not, I don't know.) Conditional convergence also seems unlikely. But it might be interesting to try to prove $\sum_{n\le X}\tan(n!)=o(X)$, or even $O(X^\epsilon)$ for some $\epsilon<1$. (I haven't thought about it, maybe this is easy, maybe not.)
Jun
22
comment Order of vanishing of an integer polynomial at a point
In the one-variable case, the condition that $f(\alpha)$ vanishes implies that $\alpha$ is algebraic over $\mathbb Q$. In your situation you're only assuming that $\alpha_1$ and $\alpha_2$ are algebraically dependent over $\mathbb Q$. If you're willing to assume that they are both algebraic over $\mathbb Q$, it might be easier to prove what you want. Or do you really need to allow the $\alpha_i$'s to be transcendental?
Jun
18
comment Roth's theorem, Lang's conjecture and beyond
Just noticed this long-ago question. I don't recall the exact reference (maybe Lang's little Diophantine Approximation book), but I thought Lang conjectured that if $f(n)$ is any positive real-valued function such that $\sum_{n\ge1} 1/f(n)$ converges, then there are only finitely many solutions to $|\alpha-\frac{p}{q}|<\frac{c}{qf(q)}$. I don't recall his discussing a possible converse as in your question (2).
Jun
11
comment Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$
... method to find lots of solutions involves the group law on elliptic curves, while the solution I described, albeit without working out the explicit equations, involves solving quadratic polynomials. So despite my admiration for the theory of elliptic curves, I think it would actually be easier to explain the quadratic equation method to, say, a high school student. OTOH, the rational curves you've found are completely explicit, which is even better.
Jun
11
comment Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$
"I am sure that Noam Elkies and Joe Silverman feel their answers are extremely simple. The following discussion is, in my humble opinion, simpler." Hi Allan, maybe that's a bit snarky? In any case, your solution and my solution are complementary. You've explained how to transform the equation into an explicit elliptic surface, which is great. I explained a very general procedure for using one solution to find lots of others. It is also very simple, since it comes down to the fact that if you know one solution of a quadratic equation, it's easy to write down the other one! So using your ...
Jun
9
answered Who needs a symmetric upper asymptotic density on the integers?
Jun
9
comment Can every genus $2$ curve be written as ramified cover of elliptic curve?
Good answer over $\mathbb C$, but the countability argument doesn't work over $\overline{\mathbb F}_p$. Nor if one wants an example defined over $\mathbb Q$.
Jun
7
comment Find all rational solutions of this diophantine-equation?
Eliminating the radicals for your second equation yields $$q^2 p^4 + \left(-4 q^3 + 4 q\right) p^3 - 2 q^2 p^2 + \left(4 q^3 - 4 q\right) p + \left(4 q^4 - 7 q^2 + 4\right)=0,$$ which is the equation of a singular plane curve. There are packages such as Sage which will desingularize and compute the genus. (If you're interested in this subject, you should learn to use one.) If the genus is $g\ge2$, then there are finitely many solutions. If $g=1$, you can find a Weierstrass model and compute the rank. If $g=0$, there are lots of solutions.
Jun
7
awarded  Enlightened
Jun
7
awarded  Nice Answer
Jun
5
answered Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$
Jun
5
comment Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$
@AlexDegtyarev Noam's comment didn't answer the question of how to find/describe all rational solutions, although it could have been given as an answer (rather than a comment) for how to find infinitely many, and possibly how to find a Zariski dense set of solutions. But in any case, characterizing rational points on K3 surfaces certainly qualifies as a research-level problem, and this seems like a nice example due to the symmetry.
Jun
4
answered N-th root of unity in N-th division field of abelian variety?
May
25
comment Are the elliptic curve discrete log problem and the elliptic curve Diffie-Hellman problem equivalent?
@ToddTrimble The wikipedia article says "Although the word acronym is widely used to refer to any abbreviation formed from initial letters some dictionaries and usage commentators define acronym to mean..." Personally, I don't see the need to distinguish between the two, but I guess that one could come up with a situation where one might want to make the distinction. Chacun a son ...
May
24
comment Are the elliptic curve discrete log problem and the elliptic curve Diffie-Hellman problem equivalent?
@Todd Is "initialization" the new term for "acronym"? :) I like it, but unfortunately initialization already has another standard meaning. (I also agree with you that on MO, one should avoid using acronyms without defining them.)