On the integers not of the form $p+2^a+2^b$

Hao Pan

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Displaying similar documents to “On the integers not of the form $p + 2^{a} + 2^{b}$ ”

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Pingzhi Yuan (2004)

Acta Arithmetica

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New infinite families of Ramanujan-type congruences modulo 9 for overpartition pairs

Ernest X. W. Xia (2015)

Colloquium Mathematicae

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Let $\bar{p p} (n)$ denote the number of overpartition pairs of n. Bringmann and Lovejoy (2008) proved that for n ≥ 0, $\bar{p p} (3 n + 2) \equiv 0 (m o d 3)$ . They also proved that there are infinitely many Ramanujan-type congruences modulo every power of odd primes for $\bar{p p} (n)$ . Recently, Chen and Lin (2012) established some Ramanujan-type identities and explicit congruences for $\bar{p p} (n)$ . Furthermore, they also constructed infinite families of congruences for $\bar{p p} (n)$ modulo 3 and 5, and two congruence relations modulo 9. In this paper, we prove several...

Some new infinite families of congruences modulo 3 for overpartitions into odd parts

Ernest X. W. Xia (2016)

Colloquium Mathematicae

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Let $p ̅_{o} (n)$ denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function $p ̅_{o} (n)$ have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for $p ̅_{o} (n)$ modulo 3. For example, we prove that for n, α ≥ 0, $p ̅_{o} (4^{α} (24 n + 17)) \equiv p ̅_{o} (4^{α} (24 n + 23)) \equiv 0 (m o d 3)$ .

Polynomial analogues of Ramanujan congruences for Han's hooklength formula

William J. Keith (2013)

Acta Arithmetica

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This article considers the eta power $\prod_{(1 - q^{k})}^{b - 1}$ . It is proved that the coefficients of $q^{n} / n!$ in this expression, as polynomials in b, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when n = 5j+4. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.

On Alternatives of Polynomial Congruences

Mariusz Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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What should be assumed about the integral polynomials $f ₁ (x), . . ., f_{k} (x)$ in order that the solvability of the congruence $f ₁ (x) f ₂ (x) \dots f_{k} (x) \equiv 0 (m o d p)$ for sufficiently large primes p implies the solvability of the equation $f ₁ (x) f ₂ (x) \dots f_{k} (x) = 0$ in integers x? We provide some explicit characterizations for the cases when $f_{j} (x)$ are binomials or have cyclic splitting fields.

On the congruences $σ (n) \equiv a (m o d n)$ and $n \equiv a (m o d φ (n))$

Carl Pomerance (1975)

Acta Arithmetica

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Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Zhi-Hong Sun (2011)

Acta Arithmetica

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Fermat numbers and integers of the form $a^{k} + a^{l} + p^{α}$

Yong-Gao Chen, Rui Feng, Nicolas Templier (2008)

Acta Arithmetica

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Density of solutions to quadratic congruences

Neha Prabhu (2017)

Czechoslovak Mathematical Journal

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A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly $k$ prime factors for $k > 1$ . Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree $n \leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^{k}$ solutions modulo $n$ . ...

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

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For k = 1,2,... let $H_{k}$ denote the harmonic number $\sum_{j = 1}^{k} 1 / j$ . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have $\sum_{k = 1}^{p - 1} (H_{k}) / (k 2^{k}) \equiv 7 / 24 p B_{p - 3} (m o d p ²)$ , $\sum_{k = 1}^{p - 1} (H_{k, 2}) / (k 2^{k}) \equiv - 3 / 8 B_{p - 3} (m o d p)$ , and $\sum_{k = 1}^{p - 1} (H ²_{k, 2 n}) / (k^{2 n}) \equiv ((\binom{6 n + 1}{2 n - 1}) + n) / (6 n + 1) p B_{p - 1 - 6 n} (m o d p ²)$ for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and $H_{k, m} : = \sum_{j = 1}^{k} 1 / (j^{m})$ .

Non-abelian $p$ -adic $L$ -functions and Eisenstein series of unitary groups – The CM method

Thanasis Bouganis (2014)

Annales de l’institut Fourier

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In this work we prove various cases of the so-called “torsion congruences” between abelian $p$ -adic $L$ -functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results...

Erratum to the paper "On the disc theorem" (Ann. Polon. Math. 55 (1991), 1-10)

Cabiria Andreian Cazacu (1992)

Annales Polonici Mathematici

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Due to a technical error, part of a sentence was omitted on the top of page 8. The first line should read: “where $f_{p}^{k}$ , $p = a_{l}$ or $b_{l}$ , means the number of folds of the covering $(δ_{k}^{''}, T |, Δ_{l}^{''})$ ending at p, i.e. covering a neighbourhood of p in $a_{l} b_{l}$ without covering p itself”.

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 square classes in generalized Fibonacci sequences

Zafer Şiar, Refik Keskin (2016)

Acta Arithmetica

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Let P and Q be nonzero integers. The generalized Fibonacci and Lucas sequences are defined respectively as follows: U₀ = 0, U₁ = 1, V₀ = 2, V₁ = P and $U_{n + 1} = P U ₙ + Q U_{n - 1}$ , $V_{n + 1} = P V ₙ + Q V_{n - 1}$ for n ≥ 1. In this paper, when w ∈ 1,2,3,6, for all odd relatively prime values of P and Q such that P ≥ 1 and P² + 4Q > 0, we determine all n and m satisfying the equation Uₙ = wUₘx². In particular, when k|P and k > 1, we solve the equations Uₙ = kx² and Uₙ = 2kx². As a result, we determine all n such that Uₙ = 6x². ...

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer $n \geq 3$ , let $u_{1}, ..., u_{n}$ be pairwise coprime integers $\geq 2$ , $𝒟$ a family of nonempty proper subsets of ${1, ..., n}$ with “enough” elements, and $ε$ a function $𝒟 \to {\pm 1}$ . Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_{i} - ε (I)$ for some $I \in 𝒟$ , but it does not divide $u_{1} \dots u_{n}$ ? We answer this question in the positive when the $u_{i}$ are prime powers and $ε$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if $ε_{0} \in {\pm 1}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k \geq 2$ and of the same level $N$ , both eigenfunctions of the Hecke operators, and both normalized so that $a_{1} (f) = a_{1} (E) = 1$ . The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\bar{ℚ}$ lying over a rational prime $p > 2$ ), the algebraic parts of the special values $L (E, χ, j)$ and $L (f, χ, j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

Associated primes, integral closures and ideal topologies

Reza Naghipour (2006)

Colloquium Mathematicae

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Let ⊆ be ideals of a Noetherian ring R, and let N be a non-zero finitely generated R-module. The set Q̅*(,N) of quintasymptotic primes of with respect to N was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set $A *_{a} (, N) : = ⋃_{n \geq 1} A s s_{R} R / {(ⁿ)}_{a}^{(N)}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ${{(ⁿ)}_{a}^{(N)} :_{R} ⟨ ⟩}_{n \geq 1}$ is finer than the topology defined by ${(ⁿ)}_{a}^{(N)}_{n \geq 1}$ if and only if $A *_{a} (, N)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover,...

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

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Consider a recurrence sequence ${(x_{k})}_{k \in ℤ}$ of integers satisfying $x_{k + n} = a_{n - 1} x_{k + n - 1} + . . . + a ₁ x_{k + 1} + a ₀ x_{k}$ , where $a ₀, a ₁, . . ., a_{n - 1} \in ℤ$ are fixed and a₀ ∈ -1,1. Assume that $x_{k} > 0$ for all sufficiently large k. If there exists k₀∈ ℤ such that $x_{k ₀} < 0$ then for each negative integer -D there exist infinitely many rational primes q such that $q | x_{k}$ for some k ∈ ℕ and (-D/q) = -1.

Some congruences involving binomial coefficients

Hui-Qin Cao, Zhi-Wei Sun (2015)

Colloquium Mathematicae

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Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that $T_{p - 1} \equiv (p / 3) 3^{p - 1} (m o d p ²)$ , where the central trinomial coefficient Tₙ is the constant term in the expansion of $(1 + x + x^{- 1}) ⁿ$ . We also prove three congruences modulo p³ conjectured by Sun, one of which is $\sum_{k = 0}^{p - 1} (\binom{p - 1}{k}) (\binom{2 k}{k}) ({(- 1)}^{k} - {(- 3)}^{- k}) \equiv (p / 3) (3^{p - 1} - 1) (m o d p ³)$ . In addition, we get some new combinatorial identities.

Displaying similar documents to “On the integers not of the form p + 2 a + 2 b ”

Displaying similar documents to “On the integers not of the form $p + 2^{a} + 2^{b}$ ”