Noam D. Elkies
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 2d comment Are the nontrivial zeros of the Riemann zeta simple? Not that this answers your question, but a multiple zero is in some sense even less likely than one would expect if the zeros were "randomly distributed" (given their average density): they seem to repel each other, so if you plot the zero spacings scaled to average 1 then their density approaches $0$ as the distance approaches $0$. [I hedge with "in some sense" because even random spacing would make the probability of a coincidence $0$. NB I'm trying to minimize confusion by using "zero" for a root of $\zeta(s)$ and "$0$" for the smallest nonnegative real number.] 2d comment Are the nontrivial zeros of the Riemann zeta simple? @DavidHansen what about elliptic curves of rank $3$ (or for that matter even $2$) and higher? Apr 24 comment On cluster points of a particular sequence Even more simply: 3, 18, 105, 612, 3567, 20790, 121173, 706248 are divisible by 3; Dividing by 3 yields 1, 6, 35, 204, 1189, 6930, 40391, 235416 = OEIS A001109 = solutions of $8x^2+1=\Box$, and square roots of square-triangular numbers. (Naturally the A106328 comments include this connection with A00109.) Apr 20 comment magma generators for unit group/ sage totally positive (and anything that's in gp is automatically accessible by Sage even if there's no native Sage way to do it) Apr 20 comment magma generators for unit group/ sage totally positive In gp you could do K=bnfinit(x^3+x^2-2*x-1); U=K.fu; S=bnfsignunit(K) to get a vector U of fundamental units and a matrix S of their real embeddings' signs. Then it just takes a bit of linear algebra mod 2 (possibly using matsolvemod) to construct a basis of totally positive units. Apr 19 comment Smooth, irreducible surface with real part containing two projective planes I noticed this error and was correcting it as you wrote. Apr 19 revised Smooth, irreducible surface with real part containing two projective planes added 608 characters in body Apr 19 answered Smooth, irreducible surface with real part containing two projective planes Apr 19 comment Equations for Elliptic Curves (Inevitably that's not far from the usual Riemann-Roch argument, but is what people actually did in the old days and sometimes still do nowadays.) Apr 19 comment Equations for Elliptic Curves Would you prefer something along these lines? We know the curve has a model $Y^2 = P(X)$ with $P$ of degree $3$ or $4$ and some rational point. Changing projective coordinates, we may assume this point has $X = \infty$. If $\deg P = 3$ then we're done. Else $\deg P = 4$ and the leading coefficient of $P$ is a square, so at infinity $P = Q^2 + R$ for some $Q,R$ with rational coefficients such that $\deg Q = 2$ and $\deg Q \leq 1$. Now write $Y=Q+x$ and get an equation quadratic in $X$ whose discriminant is cubic in $x$. This gives a birational map from $Y^2 = P(X)$ to $y^2 = cubic(x)$. Apr 18 comment What does “game theory” cover and how should it be called? Combinatorial game theory can also be summarized as an extension of the Sprague-Grundy theory en.wikipedia.org/wiki/Sprague-Grundy_theorem of impartial games ( = generalized Nim ) to games that may be "partial" in the sense that the two opponents needn't always have the same move sets. Apr 17 comment On the parity of $[x^n]$ ...(as well as Max Alekseyev's $(3+\sqrt{17})/2$ example). Apr 17 comment On the parity of $[x^n]$ It was a guess when I wrote it (and the comment above), but meanwhile I thought about it a bit more and I think we can construct $x$ by intersecting an infinite sequence of nested intervals as long as the first interval $[x_1, x_1+1)$ has $x_1 \geq 4$ (so $4$ or $5$ will do). The $n$-th interval has the form $[x_n, (x_n^n+1)^{1/n})$ for some integer $x_n$, and that puts $x^{n+1}$ in an interval of length at least 4 so its intersection with $\{X\in{\bf R}:\lfloor X\rfloor\equiv b_{n+1} \bmod 2\}$ must contain a semiopen interval of length $1$. I should update my answer to incorporate this... Apr 16 comment Is there a formula for the number of elements in $S_n$ having length $k$ with respect to the generators taken to be the transpositions? I see that "Mahonian" = in honor of MacMahon; is that a common formation? (Cf. Descartes → cartesian, but Desargues → Desarguesian [though "Arguesian" is occasionally seen too].) Apr 10 comment A long-lasting conjecture of Pólya & Szegő Is the regular $n$-gon even proved to be a local minimum for $\lambda_1$? The survey you cite doesn't cite such a result in the "case of polygons" sections (3.2, page 5). Apr 1 comment Examples of math hoaxes/interesting jokes published on April Fool's day? That's actually not far (even in many subsidiary details) from yet another item in Martin Gardner's famous April column that already accounts for two answers here. Mar 30 awarded Enlightened Mar 30 awarded Nice Answer Mar 30 revised On the parity of $[x^n]$ Add the explicit example with a (totally real, discriminant 148) cubic irrationality. Mar 30 comment On the parity of $[x^n]$ @ARupinski Yes. To get a prime between $N$ and $N+H$, we need $H$ to be some power of $N$ (it should really be a power of $\log N$ but we can't prove anything like that), which is why Mills needs double exponentials. For an integer of given parity, $H=2$ is enough, so plain $[x^n]$ probably suffices.