Number Theory Seminar: The polynomials X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2 also capture their primes

Event Type

Seminar/Symposium

Sponsor

n/a

Virtual

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Date

Feb 22, 2022 11:00 am

Speaker

Jori Merikoski (Oxford)

Contact

Kevin Ford

E-Mail

ford126@illinois.edu

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Abstract: We show that there are infinitely many primes of the form X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2.

Our work builds on the famous Friedlander-Iwaniec result on primes of the form X^2+Y^4.

More precisely, Friedlander and Iwaniec obtained an asymptotic formula for the number of primes of this form.

For the argument we need to estimate Type II sums, which is achieved by an application of the Weil bound,

both for point-counting and for exponential sums over curves. The type II information we get is too narrow

for an asymptotic formula, but we can apply Harman's sieve method to establish a lower bound of the

correct order of magnitude for the number of primes of the form X^2+(Y^2+1)^2 and X^2 + (Y^3+Z^3)^2.

Zoom: https://illinois.zoom.us/j/84027824197?pwd=TzkyUGRpeDZIcnVxTnpvVEVyLytXUT09

Meeting ID: 840 2782 4197

Password: 271403

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