bio  website  math.harvard.edu/~elkies 

location  Harvard University, Cambridge, MA 02138  
age  48  
visits  member for  3 years, 9 months 
seen  4 mins ago  
stats  profile views  22,364 
[Not sure yet what to use this space for...]
3h

comment 
Question regarding a theorem of Erdos and Renyi on $B_2(g)$ sequence
Looks like this is just counting in two ways solutions of $k=s_1+s_2$ with $s_1 \leq s_2 \leq n$ and $k \leq 2n$ to get $S(n)^2 \leq 4gn$. 
2d

awarded  Enlightened 
Jan 25 
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Identity for Power Series and Binomial Coefficients
You're welcome. For other $j$ there doesn't seem to be such a nice formula, except in the roughlycomplementary case of $j=k1$. Otherwise the powerseries coefficients, written as polynomials in $N$, are generally irreducible once you remove the factor of $N$. (For $j=1$ they factor completely, which soon led me to surmise the formula above.) 
Jan 25 
comment 
Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
Thanks. Basically I just used the routine lindep in gp. Once one has guessed such an identity numerically it's usually not hard to prove. Here $f(n)$ is odd and 7antiperiodic, so $\sum_{n=1}^\infty f(n)/n = 0$ comes down to a linear relation among the cosecants of multiples of $\pi/7$, namely $$ \frac1{\sin \pi/7} = \frac1{\sin 2\pi/7} + \frac1{\sin 3\pi/7}. $$ 
Jan 25 
answered  Identity for Power Series and Binomial Coefficients 
Jan 25 
comment 
Identity for Power Series and Binomial Coefficients
What's the connection with representation theory? 
Jan 25 
awarded  Nice Answer 
Jan 24 
answered  Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$. 
Jan 20 
comment 
Number of critical points of smooth functions on $S^1$
There is a generalization in $T^1$, though: if $u$ is orthogonal to $\cos n\theta$ and $\sin n\theta$ for each positive $n < d$ then $u$ has at least $2d$ critical points. (Your question is the case $d=2$; both John Pardon's proof and the one I gave readily generalize to arbitrary $d$.) 
Jan 20 
comment 
Number of critical points of smooth functions on $S^1$
Sorry, I don't see a direct generalization to higherdimensional spheres. 
Jan 20 
revised 
Number of critical points of smooth functions on $S^1$
corrected typo in second display (second integral) 
Jan 20 
answered  Number of critical points of smooth functions on $S^1$ 
Jan 13 
awarded  Nice Answer 
Jan 12 
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Reference for hyperelliptic curves
Thanks. I must have learned this from Joe Harris. 
Jan 11 
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Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
(About the older comment: sure, but at some point it becomes infeasible to factor bivariate polynomials whose degree grows with $km$ while it's still possible to compute resultants of degrees $k$ and $m$.) 
Jan 11 
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Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
That's covered by the $(u',v')$ test I gave. When $m$ is even you can't take $u,v$ to be the $X,Y$ of the classical Weierstrass equation because then the map $(u,v) \mapsto (u^k,v^m)$ is not generically $1:1$ (try $(u',v') = (u,v)$). But $(X,Y+1)$ works. (Also for $Y^2=X^3+2$ if $3 \mid k$ you must tweak $u$ to avoid triplication on the $X$ side.) 
Jan 10 
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Reference for hyperelliptic curves
$\ldots$, $\kappa$ is a $2:1$ map to a rational normal curve, and if $\lambda$ preserves that curve $\kappa(C)$ then it comes from a fractional linear transformation of ${\bf P}^1$, etc. 
Jan 10 
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Reference for hyperelliptic curves
More generally: A curve $C$ of genus $g>1$ has a canonical map $\kappa: C \rightarrow {\bf P}^{g1}$. Since it's canonical, for any automorphism $\alpha: C \rightarrow C$ the composite map $\kappa \alpha: C \rightarrow {\bf P}^{g1}$ must be the same as $\kappa$ up to some linear automorphism $\lambda$ of ${\bf P}^{g1}$, so $\kappa\alpha = \lambda\kappa$. For example, the automorphisms of a plane quartic are exactly the linear automorphisms of the plane that preserve the quartic. In our hyperelliptic setting$\ldots$ [cont'd because of 600character limit] 
Jan 10 
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Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
You're welcome. I think the resultant is right. I just edited to add the gp code. I also checked that a few multiples of $(1,1)$ yield rational multiples whose coordinates are the expected $11$th and $7$th powers. 
Jan 10 
revised 
Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?
add requirement that u,v have no zero or pole at the test point, and a line of gp code; fix typos; 