• Corpus ID: 239024326

On improving a Schur-type theorem in shifted primes

@inproceedings{Wang2021OnIA,
  title={On improving a Schur-type theorem in shifted primes},
  author={Ruoyi Wang},
  year={2021}
}
We show that if N ≥ exp(exp(exp(k))), then any k-colouring of the primes that are less than N contains a monochromatic solution to p1 − p2 = p3 − 1. 
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References

SHOWING 1-10 OF 15 REFERENCES
A SCHUR-TYPE ADDITION THEOREM FOR PRIMES
Abstract Suppose that all primes are colored with k colors. Then there exist monochromatic primes p 1 , p 2 , p 3 such that p 1 + p 2 = p 3 + 1 .
On a theorem of Sárközy for difference sets and shifted primes
Abstract We show that if the difference of two elements of a set A ⊆ [ N ] is never one less than a prime number, then | A | = O ( N exp ⁡ ( − c ( log ⁡ N ) 1 / 3 ) ) for some absolute constant c > 0…
Difference sets and the primes
Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.
Roth's theorem in the primes
We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood…
Some of My Favorite Problems in Ramsey Theory
TLDR
In this brief note, a variety of problems from Ramsey theory on which I would like to see progress made are described and I am offering modest rewards for most of these problems.
On sets of natural numbers whose difference set contains no squares
We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J/ n{l,...,«} is O(l/log«) c »), cn->-oo. This improves previous…
Linear equations in primes
Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation…
Schur Number Five
TLDR
A proof of the solution of a century-old problem, known as Schur Number Five, is constructed and validated using a formally verified proof checker, demonstrating that any result by satisfiability solvers can now be validated using highly trustworthy systems.
Analytic Number Theory
Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large…
On monochromatic solutions to $$x-y=z^2$$
For $$k \in \mathbb {N}$$ , write S(k) for the largest natural number such that there is a k-colouring of $$\{1, \ldots ,S(k)\}$$ with no monochromatic solution to $$x-y=z^2$$ . That S(k)…
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