Most odd-degree binary forms fail to primitively represent a square

Let F be a separable integral binary form of odd degree N ≥ 5 . A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree- N superelliptic equation y² = F(x,z) has finitely many primitive integer solutions. In this paper, we consider the family FN(f₀) of degree- N superelliptic equations with fixed leading coefficient f₀ ∈ Z ± Z² , ordered by height. For every sufficiently large N , we prove that among equations in the family FN(f₀) , more than $74.9\,%$ are insoluble, and more than $71.8\,%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,%$ and $96.7\,%$ , respectively, when f₀ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over Q have no rational points.