Generalizing a theorem of Schur

Lenny Jones

Displaying similar documents to “Generalizing a theorem of Schur”

Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh (2015)

Acta Arithmetica

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In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $(x) = {g x + h_{j}}_{j = 1}^{k}$ of linear forms in ℤ[x], the set $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $(n) = {g n + h_{j}}_{j = 1}^{k}$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps $δ_{1}, . . ., δ_{m}$ form an increasing...

Infinitude of Wilson primes for $_{q} [t]$

Jim Sauerberg, Linghsueh Shu, Dinesh S. Thakur, George Todd (2013)

Acta Arithmetica

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On a divisibility problem

Shichun Yang, Florian Luca, Alain Togbé (2019)

Mathematica Bohemica

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Let $p_{1}, p_{2}, \dots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k \geq 5$ , then $(p_{k + 1} - 1)! ∣ (\frac{1}{2} (p_{k + 1} - 1))! p_{k}!,$ which improves a previous result of the second author.

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas (2006)

Acta Mathematica Universitatis Ostraviensis

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We are interested whether there is a nonnegative integer $u_{0}$ and an infinite sequence of digits $u_{1}, u_{2}, u_{3}, \dots$ in base $b$ such that the numbers $u_{0} b^{n} + u_{1} b^{n - 1} + \dots + u_{n - 1} b + u_{n},$ where $n = 0, 1, 2, \dots,$ are all prime or at least do not have prime divisors in a finite set of prime numbers $S .$ If any such sequence contains infinitely many elements divisible by at least one prime number $p \in S,$ then we call the set $S$ unavoidable with respect to $b$ . It was proved earlier that unavoidable sets in base $b$ exist if $b \in {2, 3, 4, 6},$ and that no unavoidable set exists in base $b = 5 .$ Now,...

On divisibility of the class number of real octic fields of a prime conductor $p = n^{4} + 16$ by $p$

Stanislav Jakubec (1994)

Archivum Mathematicum

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The aim of this paper is to prove the following Theorem Theorem Let $K$ be an octic subfield of the field $Q (ζ_{p} + ζ_{p}^{- 1})$ and let $p = n^{4} + 16$ be prime. Then $p$ divides $h_{K}$ if and only if $p$ divides $B_{j}$ for some $j = \frac{p - 1}{8}$ , $3 \frac{p - 1}{8}$ , $5 \frac{p - 1}{8}$ , $7 \frac{p - 1}{8}$ .

Composite positive integers whose sum of prime factors is prime

Florian Luca, Damon Moodley (2020)

Archivum Mathematicum

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In this note, we show that the counting function of the number of composite positive integers $n \leq x$ such that $β (n) = \sum_{p ∣ n} p$ is a prime is of order of magnitude at least $x / {(log x)}^{3}$ and at most $x / log x$ .

Another version of cosupport in $D (R)$

Junquan Qin, Xiao Yan Yang (2023)

Czechoslovak Mathematical Journal

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The goal of the article is to develop a theory dual to that of support in the derived category $D (R)$ . This is done by introducing ‘big’ and ‘small’ cosupport for complexes that are different from the cosupport in D. J. Benson, S. B. Iyengar, H. Krause (2012). We give some properties for cosupport that are similar, or rather dual, to those of support for complexes, study some relations between ‘big’ and ‘small’ cosupport and give some comparisons of support and cosupport. Finally, we investigate...

On prime values of reducible quadratic polynomials

W. Narkiewicz, T. Pezda (2002)

Colloquium Mathematicae

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It is shown that Dickson’s Conjecture about primes in linear polynomials implies that if f is a reducible quadratic polynomial with integral coefficients and non-zero discriminant then for every r there exists an integer $N_{r}$ such that the polynomial $f (X) / N_{r}$ represents at least r distinct primes.