Effective \(S\)-unit equations beyond three terms: Newman's conjecture (Q6595597)

Let \(K\) be a number field and \(S\) a finite set of places of \(K\), containing the infinite places \(S_\infty\). It is well-known that \(u\) is called \(S\)-unit if \(||u ||_\nu\) for all places \(\nu\) of \(K\) lying outside \(S\). Now consider the following equation: \N\[\Na_1 u_1 +a_2 u_2 + \dots a_n u_n \N\]\Nwhere \(\gcd(a_1 u_1,a_2 u_2, \dots, a_n u_n)=1\), \(a_i\) are fixed nonzero elements of a number field \(K\) and the \(u_i\) are \(S\)-units of \(K\). This equation is called the \(S\)-unit equation. In the paper under review, the authors deal with the case \(n=5\) with restriction \(|S|\leq 3\) and show how to effectively solve 5-term \(S\)-unit equations when the set of primes \(S\) has cardinality at most 3, and use this to provide an explicit answer to an old question of D. J. Newman on representations of integers as sums of \(S\)-units. The proof is based on a pigeonhole argument and linear forms in logarithms but they also use a new technique called ''matching''.