3 papers
math.NT2025
Primes represented by shifted quadratic forms: on primitivity and congruence classes
Elena Fuchs, Catherine Hsu, James Rickards +2
We prove lower bounds of the form for the number of primes up to primitively represented by a shifted positive definite integral binary quadratic form, a…
math.NT2024
Prime and thickened prime components in Apollonian circle packings
Holley Friedlander, Elena Fuchs, Piper Harris +5
Inspired by a question of Sarnak, we introduce the notion of a prime component in an Apollonian circle packing: a maximal tangency-connected subset having all prime curvatures. We…
math.NT2023
Bounding Lifts of Markoff Triples mod
Elisa Bellah, Siran Chen, Elena Fuchs +1
In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo solutions to the equation $X_1^2+X_2^2+…