2025/07/09 by Breuer, Florian, Punch, James
#11R16 #11R27 #11R45 #11Y40 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R11 #Secondary 11R29
paper · doi:10.48550/arxiv.2507.06579
We investigate Eisenstein discriminants, which are squarefree integers d ≡ 5 \pmod8 such that the fundamental unit εd of the real quadratic field K=ℚ(√(d)) satisfies εd ≡ 1 \pmod2OK. These discriminants are related to a classical question of Eisenstein and have connections to the class groups of orders in quadratic fields as well as to real cubic fields. We present numerical computations of Eisenstein discriminants up to 1011, suggesting that their counting function up to x is approximated by πE(x) ≈ (1)/(3π2)x - 0.024x5/6. This supports a conjecture of Stevenhagen while revealing a surprising secondary term, which is similar to (but subtly different from) the secondary term in the counting function of real cubic fields. We include technical details of our computation method, which uses a modified infrastructure approach implemented on GPUs.