On congruent primes and class numbers of imaginary quadratic fields

Nils Bruin; Brett Hemenway

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Displaying similar documents to “On congruent primes and class numbers of imaginary quadratic fields”

Generalizing a theorem of Schur

Lenny Jones (2014)

Czechoslovak Mathematical Journal

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In a letter written to Landau in 1935, Schur stated that for any integer $t > 2$ , there are primes $p_{1} < p_{2} < \dots < p_{t}$ such that $p_{1} + p_{2} > p_{t}$ . In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers $t \geq k \geq 1$ and real $ϵ > 0$ , there exist primes $p_{1} < p_{2} < \dots < p_{t}$ such that $p_{1} + p_{2} + \dots + p_{k} > (k - ϵ) p_{t} .$

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

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

Acta Arithmetica

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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...

On the Brun-Titchmarsh theorem

James Maynard (2013)

Acta Arithmetica

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The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size $x / (q^{1 / 2} ϕ (q))$ when logx/logq ≥ 8 and...

Carmichael numbers composed of primes from a Beatty sequence

William D. Banks, Aaron M. Yeager (2011)

Colloquium Mathematicae

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Let α,β ∈ ℝ be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence $ℬ_{α, β} = {(⌊ α n + β ⌋)}_{n = 1}^{\infty}$ . We conjecture that the same result holds true when α is an irrational number of infinite type.

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.

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,...

A note on primes of the form p=aq² + 1

Kaisa Matomäki (2009)

Acta Arithmetica

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On a ternary Diophantine problem with mixed powers of primes

Alessandro Languasco, Alessandro Zaccagnini (2013)

Acta Arithmetica

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Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality $| λ ₁ p ₁ + λ ₂ p ² ₂ + λ ₃ p ₃^{k} + ϖ | \leq {(m a x_{j} p_{j})}^{- (33 - 29 k) / (72 k) + ε}$ has infinitely many solutions in prime variables p₁, p₂, p₃.

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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The relative class number $H_{d} (f)$ of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of $_{f}$ and $_{K}$ , where $_{K}$ denotes the ring of integers of K and $_{f}$ is the order of conductor f given by $ℤ + f_{K}$ . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

The binary Goldbach conjecture with primes in arithmetic progressions with large modulus

Claus Bauer, Yonghui Wang (2013)

Acta Arithmetica

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It is proved that for almost all prime numbers $k \leq N^{1 / 4 - ϵ}$ , any fixed integer b₂, (b₂,k) = 1, and almost all integers b₁, 1 ≤ b₁ ≤ k, (b₁,k) = 1, almost all integers n satisfying n ≡ b₁ + b₂ (mod k) can be written as the sum of two primes p₁ and p₂ satisfying $p_{i} \equiv b_{i} (m o d k)$ , i = 1,2. For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.

On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we consider the asymptotic formula for the number of the solutions of the equation $p_{1} + p_{2} + p_{3} = N$ where $N$ is an odd integer and the unknowns $p_{i}$ are prime numbers of the form $p_{i} = [n^{1 / γ_{i}}]$ . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case $γ_{1} = γ_{2} = γ_{3} = γ$ our theorem implies that every sufficiently large odd integer $N$ may be written as the sum of three Piatetski-Shapiro primes of type $γ$ for $50 / 53$ < $γ$ < $1$ . ...

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex...

Prime numbers with Beatty sequences

William D. Banks, Igor E. Shparlinski (2009)

Colloquium Mathematicae

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A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we...