Yamagishi, Shuntaro Diophantine equations in semiprimes. (English) Zbl 1442.11058 Discrete Anal. 2019, Paper No. 17, 21 p. (2019). 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. Cited in 3 Documents MSC: 11D45 Counting solutions of Diophantine equations 11D72 Diophantine equations in many variables 11P55 Applications of the Hardy-Littlewood method Keywords:Hardy-Littlewood circle method; Diophantine euqations; almost prime Citations:Zbl 0103.03102; Zbl 1428.11066 PDFBibTeX XMLCite \textit{S. Yamagishi}, Discrete Anal. 2019, Paper No. 17, 21 p. (2019; Zbl 1442.11058) Full Text: DOI arXiv References: [1] B. J. Birch,Forms in many variables. Proc. Roy. Soc. Ser. A 265 1961/1962, 245-263.1,2 · Zbl 0103.03102 [2] J. Bourgain, A. Gamburd and P. Sarnak,Affine linear sieve, expanders, and sum-product. Invent. Math. 179 (2010), no. 3, 559-644.2 · Zbl 1239.11103 [3] J. R. Chen,On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16 (1973), 157-176.1 · Zbl 0319.10056 [4] B. Cook and Á. Magyar,Diophantine equations in the primes. Invent. Math. 198 (2014), 701-737. 2,3,10,11,12 · Zbl 1360.11063 [5] B. Green and T. Tao,The primes contain arbitrarily long arithmetic progressions. Ann. of Math. (2) 167 (2008), no. 2, 481-547.1 · Zbl 1191.11025 [6] B. Green and T. Tao,Linear equations in primes. Ann. of Math. (2) 171 (2010), no. 3, 1753-1850.1 · Zbl 1242.11071 [7] B. Green and T. Tao,The Möbius function is asymptotically orthogonal to nilsequences. Ann. of Math. 175 (2012), 541-566.1 · Zbl 1347.37019 [8] B. Green, T. Tao and T. Ziegler,An inverse theorem for the Gowers Us+1[N]norm. Ann. of Math. 176 (2012), no. 2, 1231-1372.1 · Zbl 1282.11007 [9] L. K. Hua,Additive theory of prime numbers. Translations of Mathematical Monographs, Vol. 13 American Mathematical Society, Providence, RI (1965).5,14 · Zbl 0192.39304 [10] A. V. Kumchev and I. D. Tolev,An invitation to additive prime number theory. Serdica Math. J. 31 (2005), no. 1-2, 1-74.2 · Zbl 1164.11339 [11] J. Liu and P. Sarnak,Integral points on quadrics in three variables whose coordinates have few prime factors. Israel J. of math. 178 (2010), 393-426.2 · Zbl 1230.11045 [12] Á. Magyar and T. Titichetrakun,Almost prime solutions of diophantine systems of high rank. Int. J. Number Theory 13 (2017), no. 6, 1491-1514.1,2,3 · Zbl 1428.11066 [13] D. Schindler,Bihomogeneous forms in many variables. J. Théorie Nombres Bordeaux 26 (2014), 483-506.3,7,8,11 · Zbl 1425.11055 [14] D. Schindler and E. Sofos,Sarnak’s saturation problem for complete intersections. Mathematika 65 (2019), no. 1, 1-56.3 · Zbl 1454.11064 [15] R. C. Vaughan and T. D. Wooley,Waring’s problem: a survey. Number theory for the millennium III, 301-340, A. K. Peters, Natick, MA, 2002.2 · Zbl 1044.11090 [16] S. Yamagishi,Prime solutions to polynomial equations in many variables and differing degrees. Forum Math. Sigma 6 (2018), e19, 89 pp.10,16 · Zbl 1454.11181 [17] L. Zhao,The quadratic form in nine prime variables. Nagoya Math. J., 223 (1) (2016), 21-65 · Zbl 1353.11101 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.