Squares of primes and powers of 2. II.

*(English)*Zbl 0997.11082In [Monatsh. Math. 128, 283-313 (1999; Zbl 0940.11047)] the present authors established a positive lower bound of the presumably correct order of magnitude for a count of the number of representations of a large number \(N \equiv 4 \bmod 8\) as a sum of four squares of primes and \(k\) powers of 2. The principal theorem of this paper holds nominally when \(k\geq 4\), but the presence of \(O\)-terms depending on \(k\) mean that the result is non-trivial only when \(k\) is sufficiently large. Subsequently, the first two authors showed in [J. Number Theory 83, 202-225, (2000; Zbl 0961.11035)] that \(k=8330\) was admissible. The counting operation was performed using suitable weights expressed in terms of the von Mangoldt function.

In the present paper a corresponding asymptotic formula is obtained when \(N \equiv 4 \bmod 24\) and \(k\) is sufficiently large. This is deduced from a corresponding mean square result, which also implies that almost all \(n \equiv 2 \bmod 24\) can be expressed as a sum of two squares of primes and a bounded number of powers of 2.

The method now used draws on the dispersion method in a manner used by P. X. Gallagher [Invent. Math. 29, 125-142 (1975; Zbl 0305.10044)] on the analogous question about sums of primes and powers of 2.

In the present paper a corresponding asymptotic formula is obtained when \(N \equiv 4 \bmod 24\) and \(k\) is sufficiently large. This is deduced from a corresponding mean square result, which also implies that almost all \(n \equiv 2 \bmod 24\) can be expressed as a sum of two squares of primes and a bounded number of powers of 2.

The method now used draws on the dispersion method in a manner used by P. X. Gallagher [Invent. Math. 29, 125-142 (1975; Zbl 0305.10044)] on the analogous question about sums of primes and powers of 2.

Reviewer: George Greaves (Cardiff)

##### MSC:

11P32 | Goldbach-type theorems; other additive questions involving primes |

11P05 | Waring’s problem and variants |

11P55 | Applications of the Hardy-Littlewood method |

PDF
BibTeX
XML
Cite

\textit{J. Liu} et al., J. Number Theory 92, No. 1, 99--116 (2002; Zbl 0997.11082)

Full Text:
DOI

##### References:

[1] | Brüdern, J.; Fouvry, E., Lagrange’s four squares theorem with almost prime variables, J. reine angew. math., 454, 59-96, (1994) · Zbl 0809.11060 |

[2] | Gallagher, P.X., Primes and powers of 2, Invent. math., 29, 125-142, (1975) · Zbl 0305.10044 |

[3] | Ghosh, A., The distribution of αp2 modulo 1, Proc. London math. soc. (3), 42, 252-269, (1981) · Zbl 0447.10035 |

[4] | Hua, L.K., Some results in the additive prime number theory, Quart. J. math. Oxford, 9, 68-80, (1938) · Zbl 0018.29404 |

[5] | Hua, L.K., Additive theory of prime numbers, (1957), Science Press · Zbl 0192.39304 |

[6] | Hua, L.K., Introduction to number theory, (1957), Science Press |

[7] | Linnik, Yu.V., Prime numbers and powers of two, Trudy mat. inst. Steklov, 38, 151-169, (1951) |

[8] | Linnik, Yu.V., Addition of prime numbers and powers of one and the same number, Mat. sb. (N.S.), 32, 3-60, (1953) |

[9] | Liu, J.Y.; Liu, M.C., Representation of even integers as squares of primes and powers of 2, J. number theory, 83, 202-225, (2000) · Zbl 0961.11035 |

[10] | Liu, J.Y.; Liu, M.C.; Zhan, T., Squares of primes and powers of 2, Mh. math., 128, 283-313, (1999) · Zbl 0940.11047 |

[11] | Rieger, G.J., Über die summe aus einem quadrat und einem primzahlquadrat, J. reine angew. math., 251, 89-100, (1968) · Zbl 0164.05004 |

[12] | Romanoff, N.P., Über einige Sätze der additiven zahlentheorie, Math. ann., 57, 668-678, (1934) · JFM 60.0131.03 |

[13] | Shields, P., Representation of integers as a sum of two hth powers of primes, Proc. London math. soc. (3), 38, 369-384, (1979) · Zbl 0397.10045 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.