Extending an Erdős result on a Romanov type problem

Ding, Yuchen

doi:10.1007/s00013-022-01737-x

Cited by 3 publications

(2 citation statements)

References 11 publications

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“…Erdős [5] himself proved this conjecture for the case a i = 2 i , which gives an affirmative answer to a question of Turán. In a former note [3], the second author proved this conjecture for the case a i | a i +1 with its quantitative form, which is a slight generalization of Erdős' result. In a subsequent note, the second author and Zhou [4] proved the conjecture for the case a i = 2 p i , where p i is the i -th prime.…”

Section: Introductionmentioning

confidence: 74%

On a conjecture of Erdős

Chen¹,

Ding²

2022

Comptes Rendus. Mathématique

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Section: Introductionmentioning

confidence: 74%

On a conjecture of Erdős

Chen¹,

Ding²

2022

Comptes Rendus. Mathématique

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“…The case a i = 2 i was proved by Erdős himself as we mentioned above (Turán's question). In a former note [5], the second author gave a slight generalization of Erdős' theorem. To be precise, suppose that A = {a 1 < a 2 < a 3 < • • •} is a set satisfying A (x) > log x and a i |a i+1 for all sufficiently large x and i, then lim sup…”

Section: Introductionmentioning

confidence: 99%

Quantitative results of the Romanov type representation functions

Chen¹,

Ding²

2022

Preprint

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Let f (n) be the number of the representations of n by the sum of a prime and a power of 2. In 1950, Erdős proved the following remarkable resultErdős conjectured f (n) = o(log n) and believed this to be rather deep. In this article, we make some progress towards this old issue by showing that lim supOur proof is based on the new achievements in the distributions of primes due to Maynard. Furthermore, the argument displayed here is actually valid for much more general settings which cover the above result as a special case.

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Quantitative results of the Romanov type representation functions

Chen

Ding

2023

The Quarterly Journal of Mathematics

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For α > 0, let$$\mathscr{A}=\{a_1 \lt a_2 \lt a_3\lt\cdots\}$$and$$\mathscr{L}=\{\ell_1, \ell_2, \ell_3,\cdots\} \quad \text{(not\ necessarily\ different)}$$be two sequences of positive integers with $\mathscr{A}(m)\gt(\log m)^\alpha $ for infinitely many positive integers m and $\ell_m\lt0.9\log\log m$ for sufficiently large integers m. Suppose further that $(\ell_i,a_i)=1$ for all i. For any n, let $f_{\mathscr{A},\mathscr{L}}(n)$ be the number of the available representations listed below$$\ell_in=p+a_i \quad \left(1\le i\le \mathscr{A}(n)\right),$$where p is a prime number. It is proved that$$\limsup_{n\to \infty } \frac{f_{\mathscr{A},\mathscr{L}}(n)}{\log\log n}\gt0,$$which covers an old result of Erdős in 1950 by taking $a_i=2^i$ and $\ell_i=1$. One key ingredient in the argument is a technical lemma established here, which illustrates how to pick out the admissible parts of an arbitrarily given set of distinct linear functions. The proof then reduces to the verifications of a hypothesis involving well-distributed sets introduced by Maynard, which of course would be the other key ingredient in the argument.

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Extending an Erdős result on a Romanov type problem

Cited by 3 publications

References 11 publications

On a conjecture of Erdős

On a conjecture of Erdős

Quantitative results of the Romanov type representation functions

Quantitative results of the Romanov type representation functions

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