The strong n-terms abc conjecture

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Let n≥3n\ge3. For non-zero integers x1,…,xnx_1,\ldots,x_n, write rad⁡(x1x2⋯xn)\operatorname{rad}(x_1x_2\cdots x_n) for the largest positive squarefree divisor of their product. The strong n-terms abc conjecture. There is a number MnM_n depending only on nn such that, given non-zero integers x1,…,xnx_1,\ldots,x_n satisfying

gcd⁡(xi,xj)=1for all i≠j,\gcd(x_i,x_j)=1\quad\text{for all }i\ne j,

and

x1+⋯+xn=0,x_1+\cdots+x_n=0,

one has

max⁡1≤j≤n∣xj∣≤rad⁡(x1x2⋯xn)Mn.\max_{1\le j\le n}|x_j|\le \operatorname{rad}(x_1x_2\cdots x_n)^{M_n}.

This is a modification of the n-terms abc conjecture discussed in Browkin's work, replacing the global gcd and nonvanishing proper-subsums conditions by pairwise coprimality and nonzero terms. Its status is not resolved in the supplied source.

References

Primary source

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

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