Browkin–Brzeziński's n-terms abc conjecture

Let n3n\ge3. For integers x1,,xnx_1,\ldots,x_n, write rad(x1x2xn)\operatorname{rad}(x_1x_2\cdots x_n) for the largest positive squarefree divisor of their product. The n-terms abc conjecture. There is a number MnM_n depending only on nn such that, given integers x1,,xnx_1,\ldots,x_n satisfying

gcd(x1,,xn)=1,x1++xn=0,\gcd(x_1,\ldots,x_n)=1,\qquad x_1+\cdots+x_n=0,

and such that no proper subsum in the second relation vanishes, one has

max1jnxjrad(x1x2xn)Mn.\max_{1\le j\le n}|x_j|\le \operatorname{rad}(x_1x_2\cdots x_n)^{M_n}.

Browkin and Brzeziński proposed this extension of the abc conjecture to nn terms. The source also records the sharper expected choice Mn=2n5+ϵM_n=2n-5+\epsilon for every ϵ>0\epsilon>0, up to finitely many exceptions; no resolution status is supplied.

Sources & referencesView supporting material

Primary source

Hector Pasten and Rocío Sepúlveda-Manzo, “On the abc and the abcd conjectures”, arXiv:2406.05083 (2024).

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