Integers represented as a sum of primes and powers of two

Heath‐Brown, D. R.; Puchta, J.-C.

doi:10.4310/ajm.2002.v6.n3.a7

Cited by 66 publications

(54 citation statements)

References 13 publications

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“…This is actually Lemma 10 of [2]. By (8.14) of [9], we can replace (41) of [2] by C 2 ≤ 1.94, and then by the proof of Lemma 9 of [2] the assertion follows. Now we prove the Theorem.…”

Section: Proof Of Theoremmentioning

confidence: 94%

Representation of odd integers as the sum of one prime, two squares of primes and powers of 2

2007

Acta Arith.

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“…This is actually Lemma 10 of [2]. By (8.14) of [9], we can replace (41) of [2] by C 2 ≤ 1.94, and then by the proof of Lemma 9 of [2] the assertion follows. Now we prove the Theorem.…”

Section: Proof Of Theoremmentioning

confidence: 94%

Representation of odd integers as the sum of one prime, two squares of primes and powers of 2

2007

Acta Arith.

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“…Linnik's approximation to Goldbach's problem [10] consists in finding an integer K such that every sufficiently large even integer can be written as the sum of two primes and K powers of two. The lowest known value for K is K = 8 unconditionally [10] and K = 7 under the generalized Riemann hypothesis ( [10,12]). The interested reader may refer to [10] for more details about the successive improvements on K.…”

Section: Introductionmentioning

confidence: 99%

An effective version of the Bombieri–Vinogradov theorem, and applications to Chen’s theorem and to sums of primes and powers of two

Habsieger

Sivak-Fischler

2010

Arch. Math.

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“…In [12] it was shown that the binary digit sum of a large prime was equally likely to be even as it was to be odd. In a slightly different direction, it was shown in [10] that all sufficiently large even numbers are the sum of two primes, together with at most 13 powers of 2.…”

Section: Remarksmentioning

confidence: 99%

A Remark on Primality Testing and Decimal Expansions

Tao

2011

J. Aust. Math. Soc.

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We show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a = 2 has already been established by Cohen and Selfridge ['Not every number is the sum or difference of two prime powers', Math. Comput. 29 (1975), 79-81] and Sun ['On integers not of the form ±p a ± q b ', Proc. Amer. Math. Soc. 128 (2000), 997-1002], using some covering congruence ideas of Erdős. Our method is slightly different, using a partially covering set of congruences followed by an application of the Selberg sieve upper bound. As a consequence, it is not always possible to test whether a number is prime from its base a expansion without reading all of its digits. We also present some slight generalisations of these results.2010 Mathematics subject classification: primary 35J10.

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Integers represented as a sum of primes and powers of two

Cited by 66 publications

References 13 publications

Representation of odd integers as the sum of one prime, two squares of primes and powers of 2

Representation of odd integers as the sum of one prime, two squares of primes and powers of 2

An effective version of the Bombieri–Vinogradov theorem, and applications to Chen’s theorem and to sums of primes and powers of two

A Remark on Primality Testing and Decimal Expansions

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