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In this paper, we investigate the existence of Sierpi´nski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer r, there exist infinitely many Sierpi´nski numbers and Riesel numbers of the form kCr. Let S(x) be the number of positive integers r satisfying 1 ≤ r ≤ x for which kCr is a Sierpi´nski number for infinitely many k. We further show that the value S(x)/x gets arbitrarily close to 1 as x tends to infinity. Generalizations to base a-Sierpi´nski numbers and base a-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers r such that kCr is simultaneously a base a-Sierpi´nski and base a-Riesel number for infinitely many k.
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This content will become publicly available on November 28, 2025
Partitio Numerorum: sums of a prime and a number of $k$-th powers
Let $$k$$ be a natural number and let $$c=2.134693\ldots$$ be the unique real solution of the equation $$2c=2+\log (5c-1)$$ in $$[1,\infty)$$. Then, when $$s\ge ck+4$$, we establish an asymptotic lower bound of the expected order of magnitude for the number of representations of a large positive integer as the sum of one prime and $$s$$ positive integral $$k$$-th powers.
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- PAR ID:
- 10638929
- Publisher / Repository:
- European Mathematical Society
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- Volume:
- ahead of print
- ISSN:
- 1435-9855
- Page Range / eLocation ID:
- 26pp
- Subject(s) / Keyword(s):
- Waring problem Goldbach problem Hardy-Littlewood method
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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