Pasten’s small derivatives conjecture with exponent η

Prove that for every real number 0<η<1, all but finitely many coprime positive-integer triples (a,b,c) satisfying a+b=c, except those whose entries form a set {1,N,q} with q prime, admit a derivation ψ supported on the primes dividing abc such that d^ψa+d^ψb=d^ψc, a and b are ψ-independent (equivalently, their arithmetic Wronskian is nonzero), and ||ψ||<c^η.

Background

The paper studies Pasten’s arithmetic derivatives, which satisfy the Leibniz rule and are additive on a selected coprime triple a+b=c. For a derivation ψ supported on the prime divisors of abc, the norm ||ψ|| is the largest absolute value of ψ on the corresponding prime generators, and ψ-independence of a and b means that the arithmetic Wronskian W^ψ=a d^ψb-b d^ψa is nonzero.

Pasten’s Wronskian inequality bounds the quality X=log c/log rad(abc) in terms of ||ψ||. Consequently, a derivation with ||ψ||<c^η would imply asymptotic bounds on the quality and on the approximation and power gains of every presentation. The conjecture excludes triples whose entries are {1,N,q} with q prime because such families can have comparatively large minimal derivation norms; the paper discusses this obstruction later in the section.