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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:
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