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2 results

DECIDABILITY AND CLASSIFICATION OF THE THEORY OF INTEGERS WITH PRIMES
ITAY KAPLAN, SAHARON SHELAH
Journal:

The Journal of Symbolic Logic / Volume 82 / Issue 3 / September 2017

Published online by Cambridge University Press:

08 September 2017, pp. 1041-1050

Print publication:

September 2017
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We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th (ℤ , +, 1, 0, Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th (ℤ , +, 0, Pr, <) which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

FACTORS OF CARMICHAEL NUMBERS AND A WEAK $k$-TUPLES CONJECTURE
Part of
- Elementary number theory
THOMAS WRIGHT
Journal:

Journal of the Australian Mathematical Society / Volume 100 / Issue 3 / June 2016

Published online by Cambridge University Press:

17 November 2015, pp. 421-429

Print publication:

June 2016
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In light of the recent work by Maynard and Tao on the Dickson $k$-tuples conjecture, we show that with a small improvement in the known bounds for this conjecture, we would be able to prove that for some fixed $R$, there are infinitely many Carmichael numbers with exactly $R$ factors for some fixed $R$. In fact, we show that there are infinitely many such $R$.