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Abstract What proportion of integers$$n \leq N$$may be expressed as$$x^2 + dy^2$$for some$$d \leq \Delta $$, with$$x,y$$integers? Writing$$\Delta = (\log N)^{\log 2} 2^{\alpha \sqrt {\log \log N}}$$for some$$\alpha \in (-\infty , \infty )$$, we show that the answer is$$\Phi (\alpha ) + o(1)$$, where$$\Phi $$is the Gaussian distribution function$$\Phi (\alpha ) = \frac {1}{\sqrt {2\pi }} \int ^{\alpha }_{-\infty } e^{-x^2/2} dx$$. A consequence of this is a phase transition: Almost none of the integers$$n \leq N$$can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 - \varepsilon }$$, but almost all of them can be represented by$$x^2 + dy^2$$with$$d \leq (\log N)^{\log 2 + \varepsilon}\kern-1.5pt$$.
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This content will become publicly available on June 1, 2026
DIGITALLY RESTRICTED SETS AND THE GOLDBACH CONJECTURE
Abstract We show that for any setDof at least two digits in a given baseb, almost all even integers taking digits only inDwhen written in basebsatisfy the Goldbach conjecture. More formally, if$$\mathcal {A}$$is the set of numbers whose digits basebare exclusively fromD, almost all elements of$$\mathcal {A}$$satisfy the Goldbach conjecture. Moreover, the number of even integers in$$\mathcal {A}$$which are less thanXand not representable as the sum of two primes is less than$$|\mathcal {A}\cap \{1,\ldots ,X\}|^{1-\delta }$$.
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- Award ID(s):
- 2001549
- PAR ID:
- 10638950
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Bulletin of the Australian Mathematical Society
- Volume:
- 111
- Issue:
- 3
- ISSN:
- 0004-9727
- Page Range / eLocation ID:
- 469 to 477
- Subject(s) / Keyword(s):
- Restricted digits Goldbach conjecture exceptional set
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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