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DIGITALLY RESTRICTED SETS AND THE GOLDBACH CONJECTURE

Part of: Additive number theory; partitions Elementary number theory

Published online by Cambridge University Press:  04 February 2025

JAMES CUMBERBATCH Open the ORCID record for JAMES CUMBERBATCH [Opens in a new window]
JAMES CUMBERBATCH*
Affiliation:
Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067, USA
*
e-mail: jcumberb@purdue.edu
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Abstract

We show that for any set D of at least two digits in a given base b, almost all even integers taking digits only in D when written in base b satisfy the Goldbach conjecture. More formally, if $\mathcal {A}$ is the set of numbers whose digits base b are exclusively from D, almost all elements of $\mathcal {A}$ satisfy the Goldbach conjecture. Moreover, the number of even integers in $\mathcal {A}$ which are less than X and not representable as the sum of two primes is less than $|\mathcal {A}\cap \{1,\ldots ,X\}|^{1-\delta }$.

Keywords

Goldbachdigitsexceptional setsdigitally restricted sets

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11A63: Radix representation; digital problems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 3 , June 2025 , pp. 469 - 477
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author’s work is supported by NSF grant DMS-2001549.

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