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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.NT2024★ 2 cited
Reciprocity obstructions in semigroup orbits in SL(2, Z)
James Rickards, Katherine E. Stange
We study orbits of semigroups of , and demonstrate reciprocity obstructions: we show that certain such orbits avoid squares, but not as a consequence of ob…