Daniel Loughran

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20172025

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math.NT2025

Fermat near misses and the integral Hilbert Property

Jessica Alessandrì, Daniel Loughran

We consider the Diophantine equation $x^{4} + y^{4} - w^{2} = n$ for $n \in Z$ , which is related to near misses for the quartic case of Fermat's Last Theorem. For certain $n$ we…

math.NT2025

Lower bounds for counting $A_{4}$ -quartic fields

Daniel Loughran, Ross Paterson

A conjecture of Malle predicts the quantity of number fields with bounded discriminant of given Galois group. We present a lower bound matching this in the case of quartic fields w…

Fermat near misses and the integral Hilbert Property

Lower bounds for counting A4​-quartic fields

Lower bounds for counting $A_{4}$ -quartic fields