Researchers propose a new “abc-type” conjecture that replaces the usual radical of an integer by a larger quantity that keeps track of how many distinct primes appear. The paper defines γ(n) as the product of the distinct primes dividing n (the usual radical) and ω(n) as the number of those primes. The new quantity is H(n) = γ(n) · (log γ(n))^{ω(n)}, and the conjecture says that for coprime positive integers a, b, c with a + b = c, the size of c is bounded by a constant times H(abc) to a small power. From this single conjecture the authors deduce several familiar-looking and new consequences about prime factors of integers in short intervals and about equations such as sums of two powers.

What the authors actually do is mostly conditional: they state Conjecture 1 with H, motivate it by analogy with a well-known polynomial theorem (the Mason–Stothers theorem), and then show how a range of number-theory results would follow if Conjecture 1 holds. A basic check is that Conjecture 1 implies the classical abc conjecture. Their main technical results include an upper bound for the function W(x,y) = sum_{j=1}^y ω(x+j), which counts distinct prime factors among y consecutive integers. Assuming the conjecture, they prove W(x,y) ≤ (1+δ)·log x / log log x for all x large enough (depending on y and δ). From that they obtain the corollary that for any fixed y, the lim sup as x→∞ of W(x,y)·log log x / log x equals 1.

Why introduce H(n)? The radical γ(n) only records which primes divide n, not how many there are or how spread out they are. By multiplying γ by (log γ)^{ω(n)}, H(n) grows faster when many distinct primes are present. Intuitively, H captures more of the factorization “complexity” of a number. The authors argue that this extra information lets one form a stronger conjecture that still respects known heuristics and produces sharper bounds on how large c can be compared with the factors a and b.