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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
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##### 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
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