Will Jagy
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 Nov 23 revised What is the lower bound for highly composite numbers? added 104 characters in body Nov 20 comment Bounds on the number of zeros of a quadratic form Keivan, I wrote a complete proof and derivation, about 25 pages, for the $2(x^2 + y^2 + z^2) - 113 (yz+zx+xy)= 0.$ If you email me (see profile) I can mail you the (Latex) pdf. Initially, I was not sure what I was looking for, so there is a section involving the adjoints of the Gram matrices that i would probably not do now. Extended notes to myself rather than a polished article. Nov 20 comment Bounds on the number of zeros of a quadratic form Meanwhile, take a look at mathoverflow.net/questions/208158/isotropic-ternary-forms/… which illustrates the FK language. Nov 19 comment Bounds on the number of zeros of a quadratic form should not have said integrally equivalent, an isotropic ternary integrally represents a multiple of $y^2 - z x.$ Nov 19 comment Bounds on the number of zeros of a quadratic form here is FK online, the whole thing. books.google.com/… Nov 19 comment Bounds on the number of zeros of a quadratic form It's pages 507-508 in FK. The result has been repeated in later summaries. In brief, an indefinite ternary form, integer coefficients, that really is isotropic over $\mathbb Q$ and therefore $\mathbb Q,$ is integrally equivalent to an integer multiple of $y^2 - zx.$ The value of this is that we can write down the primitive null vectors of this latter form, $(u^2, uv, v^2)$ up to $\pm.$ Nov 19 comment Bounds on the number of zeros of a quadratic form Alright. I have an appointment. Dimension 3 is rather special, in that we can parametrize all solutions with a finite number of recipes, similar to the formula for primitive Pythagorean triples; in your example, this follows from simple stereographic projection, in general, it comes from a fact in Fricke and Klein (1897). Meanwhile, dimension 4 has this problem: arxiv.org/abs/1205.4416 Nov 19 comment Bounds on the number of zeros of a quadratic form I think you had better include some examples with, say, indefinite binary forms. There are some results, heavily dependent on dimension, but I'm not sure what you want to do with the $\pmod m$ or why you want to do so. Nov 18 comment Proving Legendre's Sum of 3 Squares Theorem via Geometry of Numbers @johnmangual, you give a link to a Pete L. Clark writeup in comment above. In that, he discusses a very pretty approach, which is also in Serre's little book: the sum of three squares is one of Pete's ADC forms, named after Aubry and Davenport-Cassels, and means that, if $x^2 + y^2 + z^2$ represents an integer with rational values for $x,y,z,$ then it also does so with integral values. Pete and I found all positive forms for which this holds, published an article. Turned out we were confirming (and slightly correcting) an existing list by G. Nebe. Nov 3 comment Performance guarantee of RLF math.stackexchange.com/questions/1511398/… Oct 31 comment Two Vinogradovs? Is one the son of the other? @AlexandreEremenko, I just found en.wikipedia.org/wiki/Andrei_Markov on your suggestion, I will see if I can find it without the Andrei; hmmm, not yet, but I bet the page for the elder mentions Vladimir. Well, as to separate articles, en.wikipedia.org/wiki/Andrey_Markov and en.wikipedia.org/wiki/Vladimir_Andreevich_Markov and Junior: en.wikipedia.org/wiki/Andrey_Markov,_Jr. Thanks Oct 31 comment Two Vinogradovs? Is one the son of the other? @AlexandreEremenko, this begins to make sense. The impressions I had were two very different topics, then a length of time that seemed to fit father and son rather than one person. So, brothers and one son. Oct 31 comment Two Vinogradovs? Is one the son of the other? @FedorPetrov, that is something. I found some biography but it was in Russian and did not clear up anything. Oct 31 comment Two Vinogradovs? Is one the son of the other? @KConrad, I was never sure, they just seemed different topics. Oct 31 comment Two Vinogradovs? Is one the son of the other? I've never figured out Markov (Markoff). Markov processes seems to be a different person from Markov Numbers in number theory. Possibly related. Oct 20 comment Product of exponents of prime factorization @Vincent, I had it on my home machine, I ran it for a few hours, but could not do anything else with the computer. I stopped it. The next day I got Felipe's complaint and let it go. The program is correct. If you need to know, try it for a time, see what happens. Oct 19 comment Product of exponents of prime factorization @Vincent, apparently not. Oct 14 comment optimal bound in diophantine representation question well, arranged in a different order Oct 14 comment optimal bound in diophantine representation question Placed as the final of three or so answers at math.stackexchange.com/questions/829228/… Oct 14 revised optimal bound in diophantine representation question added 228 characters in body