×

zbMATH — the first resource for mathematics

Representation of even integers as sums of squares of primes and powers of 2. (English) Zbl 0961.11035
It has been shown by J. Liu, M.-C. Liu and T. Zhan [Monatsh. Math. 128, 283-313 (1999; Zbl 0940.11047)] that there is an integer \(k\) such that every sufficiently large even number may be written in the form \[ p^2_1+p^2_2+ p^2_3+p^2_4+ 2^{e_1}+ \cdots+ 2^{e_k}. \] Here it is shown that one may be take \(k=8330\). There are a couple of significant departures from the work of Liu, Liu and Zhan. Most notably, one has to estimate the number of solutions of the equation \[ n=p^2_1+ p^2_2- p^2_3- p^2_4 \] for a given non-zero integer \(n\), and the present paper uses the ‘vector sieve’ of J. Brüdern and E. Fouvry [J. Reine Angew. Math. 454, 59-96 (1994; Zbl 0809.11060)]. Although this is a clear improvement, a more direct approach is clearly desirable.

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11N36 Applications of sieve methods
11P55 Applications of the Hardy-Littlewood method
PDF BibTeX XML Cite
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. (ser. 3), 42, 252-269, (1981) · Zbl 0447.10035
[4] Greaves, G, On the representation of a number in the form x2+y2+p2+q2 where p and q are odd primes, Acta arith., 29, 257-274, (1976) · Zbl 0283.10030
[5] Halberstam, H; Richert, H.-E, Sieve methods, (1974), Academic Press London
[6] Hua, L.K, Some results in the additive prime number theory, Quart. J. math. (Oxford), 9, 68-80, (1938) · Zbl 0018.29404
[7] Hua, L.K, Introduction to number theory, (1957), Science Press Beijing
[8] Iwaniec, H, Rosser’s sieve, Acta arith., 36, 171-202, (1980) · Zbl 0435.10029
[9] Iwaniec, H, A new form of the error term in the linear sieve, Acta arith., 37, 307-320, (1980) · Zbl 0444.10038
[10] Kovalchik, F.B, Some analogies of the hardy – littlewood equation, Zap nauch. sem. leningrad. otdel. mat. inst. Steklov, 116, 86-95, (1982)
[11] Linnik, Yu.V, Prime numbers and powers of two, Trudy mat. inst. Steklov, 38, 151-169, (1951)
[12] Linnik, Yu.V, Addition of prime numbers and powers of one and the same number, Mat. sb. (N.S.), 32, 3-60, (1953)
[13] J. Y. Liu, and, M. C. Liu, The exceptional set in four prime squares problem, Illinois J. Math, to appear. · Zbl 0942.11044
[14] Liu, J.Y; Liu, M.C; Wang, T.Z, The number of powers of 2 in a representation of large even integers (I), Sci. China ser. A, 41, 386-398, (1998) · Zbl 1029.11049
[15] Liu, J.Y; Liu, M.C; Wang, T.Z, The number of powers of 2 in a representation of large even integers (II), Sci. China ser. A, 41, 1255-1271, (1998) · Zbl 0924.11086
[16] Liu, J.Y; Liu, M.C; Wang, T.Z, On the almost Goldbach problem of linnik, Journal de théorie des nombres de Bordeaux, 11, 133-147, (1999) · Zbl 0979.11051
[17] Liu, J.Y; Liu, M.C; Zhan, T, Squares of primes and powers of two, Monatsh. math., 128, 283-313, (1999) · Zbl 0940.11047
[18] Plaksin, V.A, An asymptotic formula for the number of solutions of a nonlinear equation for prime numbers, Math. USSR izv., 18, 275-348, (1982) · Zbl 0482.10045
[19] Prachar, K, Primzahlverteilung, (1957), Springer-Verlag Berlin
[20] Rieger, G.J, Über die summe aus einem quadrat und einem primzahlquadrat, J. reine angew. math., 251, 89-100, (1968) · Zbl 0164.05004
[21] Shields, P, Some applications of sieve methods in number theory, (1979), University of Wales
[22] Vinogradov, A.I, On an “almost binary” problem, Izv. akad. nauk. SSSR ser. mat., 20, 713-750, (1956) · Zbl 0072.27001
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.