On improving a Schur-type theorem in shifted primes
نویسندگان
چکیده
We show that if $$N \geq {\rm exp}({\rm exp} (k^{O(1)})))$$ , then any k-colouring of the primes are less than N contains a monochromatic solution to $$p_1 - p_2 = p_3 -1$$ .
منابع مشابه
A Schur-type Theorem for Primes
Thus if all primes are colored with k colors, then there exist arbitrarily long monochromatic arithmetic progressions. This is a van der Waerden-type [9] theorem for primes. (The well-known van der Waerden theorem states that for any m-coloring of all positive integers, there exist arbitrarily long monochromatic arithmetic progressions.) On the other hand, Schur’s theorem [7] is another importa...
متن کاملA Schur-type Addition Theorem for Primes
Thus if all primes are colored with k colors, then there exist arbitrarily long monochromatic arithmetic progressions. This is a van der Waerden-type [9] theorem for primes. (The well-known van der Waerden theorem states that for any k-coloring of all positive integers, there exist arbitrarily long monochromatic arithmetic progressions.) On the other hand, Schur’s theorem [7] is another famous ...
متن کاملA Limit Theorem for Shifted Schur Measures
To each partition λ = (λ1, λ2, . . .) with distinct parts we assign the probability Qλ(x)Pλ(y)/Z where Qλ and Pλ are the SchurQ-functions and Z is a normalization constant. This measure, which we call the shifted Schur measure, is analogous to the much-studied Schur measure. For the specialization of the first m coordinates of x and the first n coordinates of y equal to α (0 < α < 1) and the re...
متن کاملAdditive Functions on Shifted Primes
Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes. A real-valued function / defined on the positive integers is additive if it satisfies f(rs) = f(r) + f(s) whenever r and s are coprime. Such functions are determined by their values on the prime-powers. For additive arithmetic function /, let Q denote the frequency...
متن کاملImproving Roth’s theorem in the primes
Let A be a subset of the primes. Let δP (N) = |{n ∈ A : n ≤ N}| |{n prime : n ≤ N}| . We prove that, if δP (N) ≥ C log log logN (log logN)1/3 for N ≥ N0, where C and N0 are absolute constants, then A ∩ [1, N ] contains a non-trivial three-term arithmetic progression. This improves on Green’s result [Gr], which needs δP (N) ≥ C s log log log log logN log log log logN .
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Mathematica Hungarica
سال: 2023
ISSN: ['0001-5954', '0236-5294', '1588-2632']
DOI: https://doi.org/10.1007/s10474-023-01310-0