Integers represented as the sum of one prime, two squares of primes and powers of

Authors:
Guangshi Lü and Haiwei Sun

Journal:
Proc. Amer. Math. Soc. **137** (2009), 1185-1191

MSC (2000):
Primary 11P32, 11P05, 11N36, 11P55

Published electronically:
September 26, 2008

MathSciNet review:
2465639

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and powers of .

**1.**P. X. Gallagher,*Primes and powers of 2*, Invent. Math.**29**(1975), no. 2, 125–142. MR**0379410****2.**Glyn Harman and Angel V. Kumchev,*On sums of squares of primes*, Math. Proc. Cambridge Philos. Soc.**140**(2006), no. 1, 1–13. MR**2197572**, 10.1017/S0305004105008819**3.**D. R. Heath-Brown and J.-C. Puchta,*Integers represented as a sum of primes and powers of two*, Asian J. Math.**6**(2002), no. 3, 535–565. MR**1946346**, 10.4310/AJM.2002.v6.n3.a7**4.**L. K. Hua, Some results in the additive prime number theory, Quart. J. Math. (Oxford), 9(1938), 68-80.**5.**Angel V. Kumchev,*On Weyl sums over primes and almost primes*, Michigan Math. J.**54**(2006), no. 2, 243–268. MR**2252758**, 10.1307/mmj/1156345592**6.**Hongze Li,*The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes*, Acta Arith.**92**(2000), no. 3, 229–237. MR**1752027****7.**Hongze Li,*The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes. II*, Acta Arith.**96**(2001), no. 4, 369–379. MR**1811879**, 10.4064/aa96-4-7**8.**Hongze Li,*Representation of odd integers as the sum of one prime, two squares of primes and powers of 2*, Acta Arith.**128**(2007), no. 3, 223–233. MR**2313991**, 10.4064/aa128-3-3**9.**Yu. V. Linnik,*Prime numbers and powers of two*, Trudy Nat. Inst. Steklov., v. 38, Trudy Nat. Inst. Steklov., v. 38, Izdat. Akad. Nauk SSSR, Moscow, 1951, pp. 152–169 (Russian). MR**0050618****10.**Yu. V. Linnik,*Addition of prime numbers with powers of one and the same number*, Mat. Sbornik N.S.**32(74)**(1953), 3–60 (Russian). MR**0059938****11.**Jianya Liu, Mingchit Liu, and Tianze Wang,*The number of powers of 2 in a representation of large even integers. II*, Sci. China Ser. A**41**(1998), no. 12, 1255–1271. MR**1681935**, 10.1007/BF02882266**12.**Jianya Liu, Ming-Chit Liu, and Tao Zhan,*Squares of primes and powers of 2*, Monatsh. Math.**128**(1999), no. 4, 283–313. MR**1726765**, 10.1007/s006050050065**13.**Jianya Liu and Ming-Chit Liu,*Representation of even integers as sums of squares of primes and powers of 2*, J. Number Theory**83**(2000), no. 2, 202–225. MR**1772613**, 10.1006/jnth.1999.2500**14.**Tao Liu,*Representation of odd integers as the sum of one prime, two squares of primes and powers of 2*, Acta Arith.**115**(2004), no. 2, 97–118. MR**2099833**, 10.4064/aa115-2-1**15.**Jianya Liu and Guangshi Lü,*Four squares of primes and 165 powers of 2*, Acta Arith.**114**(2004), no. 1, 55–70. MR**2067872**, 10.4064/aa114-1-4**16.**J. Pintz and I. Z. Ruzsa,*On Linnik’s approximation to Goldbach’s problem. I*, Acta Arith.**109**(2003), no. 2, 169–194. MR**1980645**, 10.4064/aa109-2-6**17.**J. Pintz,*A note on Romanov’s constant*, Acta Math. Hungar.**112**(2006), no. 1-2, 1–14. MR**2251126**, 10.1007/s10474-006-0060-6**18.**Xiumin Ren,*On exponential sums over primes and application in Waring-Goldbach problem*, Sci. China Ser. A**48**(2005), no. 6, 785–797. MR**2158973**, 10.1360/03ys0341**19.**Ming Qiang Wang and Xian Meng Meng,*The exceptional set in the two prime squares and a prime problem*, Acta Math. Sin. (Engl. Ser.)**22**(2006), no. 5, 1329–1342. MR**2251394**, 10.1007/s10114-005-0701-7**20.**Tianze Wang,*On Linnik’s almost Goldbach theorem*, Sci. China Ser. A**42**(1999), no. 11, 1155–1172. MR**1749863**, 10.1007/BF02875983**21.**J. Wu,*Chen’s double sieve, Goldbach’s conjecture and the twin prime problem*, Acta Arith.**114**(2004), no. 3, 215–273. MR**2071082**, 10.4064/aa114-3-2

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
11P32,
11P05,
11N36,
11P55

Retrieve articles in all journals with MSC (2000): 11P32, 11P05, 11N36, 11P55

Additional Information

**Guangshi Lü**

Affiliation:
Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

Email:
gslv@sdu.edu.cn

**Haiwei Sun**

Affiliation:
Department of Mathematics, Shandong University, Jinan, Shandong, 250100, People’s Republic of China

DOI:
https://doi.org/10.1090/S0002-9939-08-09603-2

Keywords:
Squares of primes,
powers of $2$,
circle method.

Received by editor(s):
January 30, 2008

Received by editor(s) in revised form:
April 4, 2008

Published electronically:
September 26, 2008

Additional Notes:
This work is supported by the National Natural Science Foundation of China (Grant No. 10701048).

Communicated by:
Ken Ono

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.