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Perfect powers with few binary digits and related Diophantine problems. (English) Zbl 1303.11084
The authors prove general results on perfect powers written as short sums of monomials. More precisely, they consider the Diophantine equations in \(a\), \(b\), \(c\) and \(y\) of the form \[ x_1^a+x_2^b+1=y^q,\quad \text{ and } \quad x_1^a+x_2^b+x_3^c+1=y^q, \] where \(x_1,x_2,x_3\) are positive integers with \(\gcd(x_1,x_2)>1\) resp. \(\gcd(x_1,x_2,x_3)>1\). They show that there is no solution in \(a,b,c,y\) when \(q\geq q_0(x_i)\). In the illustrative case when \(x_1=x_2=x\), it is shown that the first equation has no solution for \(q\) being larger than an effectively computable absolute constant (under the condition that \(q\) does not divide \(\phi(x)\)). The methods rely on estimates for linear forms in two logarithms, both in Archimedean and in non-Archimedean contexts.
In a follow-up paper [Mathematika 60, No. 1, 66–84 (2014; Zbl 1372.11009)], the first two authors completely solved the first equation in the case of \(x_1=x_2=x\). Some other explicit examples for both equations can be found in complementary work by the authors [Math. Proc. Camb. Philos. Soc. 153, No. 3, 525–540 (2012; Zbl 1291.11016)].

MSC:
11J86 Linear forms in logarithms; Baker’s method
11A63 Radix representation; digital problems
11D61 Exponential Diophantine equations
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