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Diophantine equations in semiprimes. (English) Zbl 1442.11058

Summary: A semiprime is a natural number which is the product of two (not necessarily distinct) prime numbers. Let \(F(x_1,\ldots,x_n)\) be a degree \(d\) homogeneous form with integer coefficients. We provide sufficient conditions, similar to those of the seminal work of B. J. Birch [Proc. R. Soc. Lond., Ser. A 265, 245–263 (1962; Zbl 0103.03102)], for which the equation \(F(x_1,\ldots,x_n)=0\) has infinitely many integer solutions with semiprime coordinates. Previously it was known, by a result of Á. Magyar and T. Titichetrakun [Int. J. Number Theory 13, No. 6, 1491–1514 (2017; Zbl 1428.11066)], that under the same hypotheses there exist infinitely many integer solutions to the equation with coordinates that have at most \(384n^{3/2}d(d+1)\) prime factors.

MSC:

11D45 Counting solutions of Diophantine equations
11D72 Diophantine equations in many variables
11P55 Applications of the Hardy-Littlewood method
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References:

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