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ON PAIRS OF LINEAR EQUATIONS IN FOUR PRIME VARIABLES AND POWERS OF TWO

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  22 March 2012

YAFANG KONG
YAFANG KONG*
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong (email: [email protected])
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Abstract

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In this paper, we consider the simultaneous representation of pairs of positive integers. We show that every pair of large positive even integers can be represented in the form of a pair of linear equations in four prime variables and k powers of two. Here, k=63 in general and k=31 under the generalised Riemann hypothesis.

Keywords

linear equationsHardy–Littlewood methodprimespowers of two

MSC classification

Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 55 - 67
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

[1]Gallagher, P. X., ‘Primes and powers of 2’, Invent. Math. 29 (1975), 125–142.CrossRefGoogle Scholar
[2]Green, B. and Tao, T., ‘Linear equations in primes’, Ann. of Math. (2) 171(3) (2010), 1753–1850.CrossRefGoogle Scholar
[3]Heath-Brown, D. R. and Puchta, J.-C., ‘Integers represented as a sum of primes and powers of two’, Asian J. Math. 6(3) (2002), 535–566.CrossRefGoogle Scholar
[4]Linnik, Y. V., ‘Addition of prime numbers with powers of one and the same number’, Mat. Sbornik N.S. 32(74) (1953), 3–60 (in Russian).Google Scholar
[5]Liu, J.-Y., Liu, M.-C and Wang, T.-Z., ‘The number of powers of 2 in a representation of large even numbers (II)’, Sci. China Ser. A 41 (1998), 1255–1271.CrossRefGoogle Scholar
[6]Liu, J.-Y., Liu, M.-C and Wang, T.-Z, ‘On the almost Goldbach problem of Linnik’, J. Theor. Nombres Bordeaux 11 (1999), 133–147.CrossRefGoogle Scholar
[7]Liu, Z.-X. and Lv, G.-S., ‘Density of two squares of primes and powers of 2’, Int. J. Number Theory 7(5) (2011), 1317–1329.CrossRefGoogle Scholar
[8]Liu, M.-C and Tsang, K.-M, ‘On pairs of linear equations in three prime variables and an application to Goldbach’s problem’, J. reine angew. Math. 399 (1989), 109–136.Google Scholar
[9]Pintz, J., ‘A note on Romannov’s constant’, Acta Math. Hungar. 112 (2006), 1–14.CrossRefGoogle Scholar
[10]Pintz, J. and Ruzsa, I. Z., ‘On Linnik’s approximation to Goldbach’s problem I’, Acta Arith. 109(2) (2003), 169–194.CrossRefGoogle Scholar
[11]Wu, J., ‘Chen’s double sieve, Goldbach’s conjecture and the twin prime problem’, Acta Arith. 114(3) (2004), 215–273.CrossRefGoogle Scholar