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 Sep 15 reviewed Leave Closed Prime divisors in a proof Sep 15 reviewed Leave Closed Constructive Mathematics and Termination Sep 15 reviewed Reviewed Prime divisors in a proof Sep 13 comment Kronecker polynomial or not? Are you assuming that $f$ is monic? Sep 13 answered primorial puzzlement Sep 12 awarded Enlightened Sep 12 awarded Nice Answer Sep 12 comment Lattice random walk under gravity Oh I see your confusion. Your numerics are fine. You get $P=0.99$ and then $E$ which is the probability you want is $P/(1+P)$ which is about $1/2$. Sorry for calling the final answer $E$ (for exact return) rather than $P$. Sep 12 comment Lattice random walk under gravity How are you evaluating the integral? Since it is singular, some care would be needed, and maybe the numerics are off? (The formula looks fine to me, but I'll see if I can put in asymptotics easily. And I imagine you didn't forget to divide by $4\pi^2$?) Sep 12 answered Lattice random walk under gravity Sep 11 awarded Good Answer Sep 11 reviewed No Action Needed Homotopies with prescribed regular values Sep 11 comment What is wrong with this deterministic algorithm efficiently generating large primes? Hi Joël, The conjecture, which is an analog of Cramer's conjecture, is of course wildly optimistic, but I think is believed to be true. The Friedlander-Granville work disproves Montgomery's conjecture by a Maier-type method, but it is still reasonable that $\psi(x;q,a) \gg \psi(x)/\phi(q)$ in a uniform range as in the Montgomery conjecture. Same story with primes in short intervals: the asymptotic formula fails for any short interval of size $(\log x)^A$ for any $A$, but still if $A>2$ we would expect a good number of primes in such intervals. Sep 11 comment Update for 2015: least prime of form nq+1, with q prime? It wasn't so precise a remark, but if you look at the Odlyzko bounds for discriminants, then if there is a small prime that splits completely then one gets an improvement of those bounds. Thus for minimal discriminants not too far from the Odlyzko bounds, one usually has that most of the small primes are not split. Sep 10 comment Update for 2015: least prime of form nq+1, with q prime? @VesselinDimitrov: Thanks for those remarks. It seems to me that the lower bounds for Mahler measures precisely corresponds to the case of large degree and small discriminant. I don't really have a good intuition for that case -- and one also knows that discriminants of number fields are small precisely when there are no primes of small norm. It would be interesting to figure out a uniform such conjecture, but I don't know offhand what to expect. Sep 10 comment Update for 2015: least prime of form nq+1, with q prime? @VesselinDimitrov: That's an interesting observation, also consistent for example with what we believe about the least quadratic non-residue. Do you have in mind that the degree of the number field is fixed and discriminant goes to infinity, where I would believe a heuristic of this type; or do you have in mind that the degree can go to infinity as well (which is the cyclotomic case, but at least the discriminant is pretty big), where I would be more cautious (e.g. with minimal discriminants)? (Dobrowolski's argument is not so fresh in my mind that I can see what you're thinking.) Sep 10 awarded Enlightened Sep 10 comment Integer solution I don't think so: for example the three variables can all lie between $p^{1/3}/10$ and $10p^{1/3}$. I think what I said is correct. (The three dimensional multiplication table problem counts $n$ of size $x$ being factored as $n_1n_2n_3$ with $n_1$, $n_2$, $n_3$ all of size $x^{1/3}$.) Sep 10 comment Integer solution That's nice, but why do you say that $p^{1/3}$ is unlikely. It seems plausible to me: by the three dimensional multiplication table problem one expects that there are about $x/(\log x)^c$ (for a suitable constant $c$) integers of size $x$ that can be factored into three factors of roughly equal size. Given that, I would expect that any interval of length $x^{\epsilon}$ around $x$ should contain such numbers (although proving this is another matter). Sep 10 awarded Nice Answer