Timeline for $Ax^2 + By^3$ representing infinitely many primes
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 15, 2020 at 10:26 | answer | added | Ptino | timeline score: 0 | |
| Feb 5, 2019 at 11:11 | comment | added | Gerry Myerson | The first 10,000 primes of the form $a^2+b^3$, $a,b>0$, are accessible via oeis.org/A066649 | |
| Feb 5, 2019 at 4:40 | comment | added | Bombyx mori | Welcome back, Reid! | |
| Feb 5, 2019 at 0:06 | answer | added | Seewoo Lee | timeline score: 4 | |
| Feb 4, 2019 at 23:17 | comment | added | Dror Speiser | If you allow $x,y$ to be rational then the parity conjecture for elliptic curves easily implies a positive proportion of primes are representable (as long as there are no simple local obstructions, like $gcd(A, B)\ne 1$). But this doesn't help much with integer solutions... | |
| Feb 4, 2019 at 22:56 | comment | added | literature-searcher | I agree with what Lucia says (that the only currently plausible attack on $x^2+y^3$ is via $x^2+y^6$ and the latter is (just) not dense enough), but also want to note that the handling of the bilinear terms in the two papers is quite different, with H-B's being a multi-dimensional large sieve (involving the cubic field), while F-I exploits the regularity of the squares. The fact that H-B can handle density 2/3 and F-I (just) cannot derives from the use of a Brun-type sieve over a short logarithmic range in the former (splicing type I and II in the other ranges), which is unavailable for F-I. | |
| Feb 4, 2019 at 22:31 | comment | added | literature-searcher | @Wojowu Motohashi's result is pretty much subsumed by Iwaniec's 1974 work (which is an application of the half-linear sieve IIRC). mathoverflow.net/questions/55384/… matwbn.icm.edu.pl/ksiazki/aa/aa24/aa2451.pdf | |
| Feb 4, 2019 at 22:25 | comment | added | literature-searcher | The density of the sequences is the most crucial aspect for the congruential sums (Type I in the parlance), but not so much so for the bilinear sums (Type II) (though there is /some/ necessary density, by Cauchy-Schwarz considerations). Indeed, in the results cited, the arithmetic of the sequence plays the larger role in bounding the bilinear sums. On the other hand, there are examples where the bilinear estimates can come about from more flexible means, e.g. as in Duke-Friedlander-Iwaniec for equidistribution of quadratic congruences modulo primes (cf Chapter 21 of Iwaniec-Kowalski). | |
| Feb 4, 2019 at 22:10 | comment | added | Wojowu | Lack of factorization over number fields surely is going to be an issue, but I have found one reference which deals with the case $x^2+y^2+1$, which is absolutely irreducible: matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1642.pdf | |
| Feb 4, 2019 at 22:08 | comment | added | Lucia | Both the Friedlander--Iwaniec and Heath--Brown theorems use that $x^3+2y^3$ and $x^2+y^4$ are specializations of norm forms in number fields. This is lacking in the $x^2 +y^3$ situation, unless one studies it in the more special case of $x^2 + y^6$. But in that case, one does not have enough control over levels of distribution to make the Friedlander--Iwaniec argument work (at least as things stand now). The $x^2 +y^3$ problem is of interest to researchers wanting to produce elliptic curves with prime conductor. People have thought about it, so far unscuccesfully. | |
| Feb 4, 2019 at 22:00 | history | asked | Reid Barton | CC BY-SA 4.0 |