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), 11851191
MSC (2000):
Primary 11P32, 11P05, 11N36, 11P55
Published electronically:
September 26, 2008
MathSciNet review:
2465639
Fulltext PDF
Abstract 
References 
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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
(52 #315)
 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 (2007b:11153), http://dx.doi.org/10.1017/S0305004105008819
 3.
D.
R. HeathBrown 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
(2003i:11146)
 4.
L. K. Hua, Some results in the additive prime number theory, Quart. J. Math. (Oxford), 9(1938), 6880.
 5.
Angel
V. Kumchev, On Weyl sums over primes and almost primes,
Michigan Math. J. 54 (2006), no. 2, 243–268. MR 2252758
(2007g:11096), http://dx.doi.org/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
(2000m:11098)
 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
(2002d:11121), http://dx.doi.org/10.4064/aa9647
 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
(2008g:11172), http://dx.doi.org/10.4064/aa12833
 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
(14,355e)
 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
(15,602j)
 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
(2000c:11164), http://dx.doi.org/10.1007/BF02882266
 12.
Jianya
Liu, MingChit
Liu, and Tao
Zhan, Squares of primes and powers of 2, Monatsh. Math.
128 (1999), no. 4, 283–313. MR 1726765
(2000k:11110), http://dx.doi.org/10.1007/s006050050065
 13.
Jianya
Liu and MingChit
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
(2001g:11156), http://dx.doi.org/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
(2005f:11227), http://dx.doi.org/10.4064/aa11521
 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
(2005f:11226), http://dx.doi.org/10.4064/aa11414
 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
(2004c:11185), http://dx.doi.org/10.4064/aa10926
 17.
J.
Pintz, A note on Romanov’s constant, Acta Math. Hungar.
112 (2006), no. 12, 1–14. MR 2251126
(2007d:11117), http://dx.doi.org/10.1007/s1047400600606
 18.
Xiumin
Ren, On exponential sums over primes and application in
WaringGoldbach problem, Sci. China Ser. A 48 (2005),
no. 6, 785–797. MR 2158973
(2006g:11167), http://dx.doi.org/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
(2007d:11118), http://dx.doi.org/10.1007/s1011400507017
 20.
Tianze
Wang, On Linnik’s almost Goldbach theorem, Sci. China
Ser. A 42 (1999), no. 11, 1155–1172. MR 1749863
(2000m:11097), http://dx.doi.org/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
(2005e:11128), http://dx.doi.org/10.4064/aa11432
 1.
 P. X. Gallagher, Primes and powers of , Invent. Math. 29(1975), 125142. MR 0379410 (52:315)
 2.
 G. Harman and A. V. Kumchev, On sums of squares of primes, Math. Cambridge Philos. Soc., 140(2006), 113. MR 2197572 (2007b:11153)
 3.
 D. R. HeathBrown and J.C. Puchta, Integers represented as a sum of primes and powers of two, Asian J. Math., 6(2002), 535565. MR 1946346 (2003i:11146)
 4.
 L. K. Hua, Some results in the additive prime number theory, Quart. J. Math. (Oxford), 9(1938), 6880.
 5.
 A.V. Kumchev, On Weyl sums over primes and almost primes, Michigan Math. J., 54(2006), 243268. MR 2252758 (2007g:11096)
 6.
 H.Z. Li, The number of powers of in a representation of large even integers by sums of such powers and of two primes, Acta Arith., 92(2000), 229237. MR 1752027 (2000m:11098)
 7.
 H.Z. Li, The number of powers of in a representation of large even integers by sums of such powers and of two primes II, Acta Arith., 96(2001), 369379. MR 1811879 (2002d:11121)
 8.
 H.Z. Li, Representation of odd integers as the sum of one prime, two squares of primes and powers of , Acta Arith., 128(2007), 223233. MR 2313991
 9.
 Yu. V. Linnik, Primes numbers and powers of two, Trudy Mat. Inst. Steklov., 38(1951), 152169 (in Russian). MR 0050618 (14:355e)
 10.
 Yu. V. Linnik, Addition of prime numbers with powers of one and the same number, Mat. Sb.(N. S.), 32(1953), 360 (in Russian). MR 0059938 (15:602j)
 11.
 J.Y. Liu, M.C. Liu and T.Z. Wang, The number of powers of in a representation of large even integers (II), Science in China Ser. A, 41(1998), 12551271. MR 1681935 (2000c:11164)
 12.
 J.Y. Liu, M.C. Liu and T. Zhan, Squares of primes and powers of , Monatsh. Math., 128(1999), 283313. MR 1726765 (2000k:11110)
 13.
 J.Y. Liu and M.C. Liu, Representation of even integers as sums of squares of primes and powers of , J. Number Theory, 83(2000), 202225. MR 1772613 (2001g:11156)
 14.
 T. Liu, Representation of odd integers as the sum of one prime, two squares of primes and powers of , Acta Arith., 115(2004), 97118. MR 2099833 (2005f:11227)
 15.
 J.Y. Liu and G.S. Lü, Four squares of primes and powers of , Acta Arith., 114(2004), 5570. MR 2067872 (2005f:11226)
 16.
 J. Pintz and I. Z. Ruzsa, On Linnik's approximation to Goldbach's problem I, Acta Arith., 109(2003), 169194. MR 1980645 (2004c:11185)
 17.
 J. Pintz, A note on Romanov's constant, Acta Math. Hungar., 112(2006), 114. MR 2251126 (2007d:11117)
 18.
 X. M. Ren, On exponential sums over primes and applications in the WaringGoldbach problems, Science in China Ser. A, 48(2005), 785797. MR 2158973 (2006g:11167)
 19.
 M.Q. Wang and X.M. Meng, The exceptional set in the two prime squares and a prime problem, Acta Mathematica Sinica (English series), 22(2006), 13291342. MR 2251394 (2007d:11118)
 20.
 T.Z. Wang, On Linnik's almost Goldbach theorem, Science in China Ser. A, 42(1999), 11551172. MR 1749863 (2000m:11097)
 21.
 J. Wu, Chen's double sieve, Goldbach's conjecture and the twin prime theorem, Acta Arith., 114(2004), 215273. MR 2071082 (2005e:11128)
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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:
http://dx.doi.org/10.1090/S0002993908096032
PII:
S 00029939(08)096032
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.
