Products of factorials modulo p

Florian Luca; Pantelimon Stănică

Displaying similar documents to “Products of factorials modulo p”

Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\operatorname{sgn}x\geq 0$ for $|x|>R$ .

Ján Andres (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ , si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

On sets which contain a qth power residue for almost all prime modules

Mariusz Ska/lba (2005)

Colloquium Mathematicae

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A classical theorem of M. Fried [2] asserts that if non-zero integers $β ₁, . . ., β_{l}$ have the property that for each prime number p there exists a quadratic residue $β_{j}$ mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].

Shifted values of the largest prime factor function and its average value in short intervals

Jean-Marie De Koninck, Imre Kátai (2016)

Colloquium Mathematicae

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We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q} (n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence ${(α s_{q} (P (n)))}_{n \in ℕ}$ is uniformly distributed modulo 1.

The largest prime factor of X³ + 2

A. J. Irving (2015)

Acta Arithmetica

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Improving on a theorem of Heath-Brown, we show that if X is sufficiently large then a positive proportion of the values n³ + 2: n ∈ (X,2X] have a prime factor larger than $X^{1 + 10^{- 52}}$ .

Positive solutions of a fourth-order differential equation with integral boundary conditions

Seshadev Padhi, John R. Graef (2023)

Mathematica Bohemica

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We study the existence of positive solutions to the fourth-order two-point boundary value problem $\{\begin{matrix} u^{''''} (t) + f (t, u (t)) = 0, & 0 < t < 1, \\ u^{'} (0) = u^{'} (1) = u^{''} (0) = 0, & u (0) = α [u], \end{matrix}$ where $α [u] = \int_{0}^{1} u (t) d A (t)$ is a Riemann-Stieltjes integral with $A \geq 0$ being a nondecreasing function of bounded variation and $f \in 𝒞 ([0, 1] \times ℝ_{+}, ℝ_{+})$ . The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

$E_{1}$ -degeneration and $d^{'} d^{''}$ -lemma

Tai-Wei Chen, Chung-I Ho, Jyh-Haur Teh (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a double complex $(A, d^{'}, d^{''})$ , we show that if it satisfies the $d^{'} d^{''}$ -lemma and the spectral sequence ${E_{r}^{p, q}}$ induced by $A$ does not degenerate at $E_{0}$ , then it degenerates at $E_{1}$ . We apply this result to prove the degeneration at $E_{1}$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d^{'} d^{''}$ -lemma.

On the non-convergence of successive approximations in the Darboux problem for the equation $z_{x y}^{''} = f (x, y, z)$

M. Kisielewicz (1977)

Colloquium Mathematicae

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On the index of an odd perfect number

Feng-Juan Chen, Yong-Gao Chen (2014)

Colloquium Mathematicae

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Suppose that N is an odd perfect number and $q^{α}$ is a prime power with $q^{α} | | N$ . Define the index $m = σ (N / q^{α}) / q^{α}$ . We prove that m cannot take the form $p^{2 u}$ , where u is a positive integer and 2u+1 is composite. We also prove that, if q is the Euler prime, then m cannot take any of the 30 forms q₁, q₁², q₁³, q₁⁴, q₁⁵, q₁⁶, q₁⁷, q₁⁸, q₁q₂, q₁²q₂, q₁³q₂, q₁⁴ q₂, q₁⁵q₂, q₁²q₂², q₁³q₂², q₁⁴q₂², q₁q₂q₃, q₁²q₂q₃, q₁³q₂q₃, q₁⁴q₂q₃, q₁²q₂²q₃, q₁²q₂²q₃², q₁q₂q₃q₄, q₁²q₂q₃q₄, q₁³q₂q₃q₄, q₁²q₂²q₃q₄, q₁q₂q₃q₄q₅, q₁²q₂q₃q₄q₅,...

Boundedness criteria for a class of second order nonlinear differential equations with delay

Daniel O. Adams, Mathew Omonigho Omeike, Idowu A. Osinuga, Biodun S. Badmus (2023)

Mathematica Bohemica

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We consider certain class of second order nonlinear nonautonomous delay differential equations of the form $a (t) x^{''} + b (t) g (x, x^{'}) + c (t) h (x (t - r)) m (x^{'}) = p (t, x, x^{'})$ and ${(a (t) x^{'})}^{'} + b (t) g (x, x^{'}) + c (t) h (x (t - r)) m (x^{'}) = p (t, x, x^{'}),$ where $a$ , $b$ , $c$ , $g$ , $h$ , $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results....

Boundedness results of solutions to the equation $x^{\prime\prime\prime}+ax^{\prime\prime}+g(x)x^{\prime}+h(x)=p(t)$ without the hypothesis $h(x)\,\text{sgn}x\geq 0$ $for|x|>R$ .

Ján Andres (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Periodic solutions of $x^{''} + f (x) x^{' 2 n} + g (x) = μ p (t)$

S. Sędziwy (1969)

Annales Polonici Mathematici

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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.

On the distribution of prime numbers which are of the form $x^{2} + y^{2} + 1$

Yoichi Motohashi (1970)

Acta Arithmetica

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

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

Acta Arithmetica

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On hyperbolic partial differential equations in Banach spaces

Bogdan Rzepecki (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Viene dimostrata l'esistenza di soluzioni del problema di Darboux per l'equazione iperbolica $z^{\prime\prime}_{xy}=f(x,y,z,Z^{\prime}_{x},z_{y})$ sul planiquarto $x\geq 0$ , $y\geq 0$ . Qui, $f$ è una funzione continua, con valori in uno spazio Banach che soddisfano alcune condizioni di regolarità espresse in termini della misura di non-compattezza $\alpha$ .

Positive solutions for a system of third-order differential equation with multi-point and integral conditions

Rochdi Jebari, Abderrahman Boukricha (2015)

Commentationes Mathematicae Universitatis Carolinae

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This paper concerns the following system of nonlinear third-order boundary value problem: $u_{i}^{'''} (t) + f_{i} (t, u_{1} (t), \dots, u_{n} (t), u_{1}^{'} (t), \dots, u_{n}^{'} (t)) = 0, 0 < t < 1, i \in {1, \dots, n}$ with the following multi-point and integral boundary conditions: $\{\begin{matrix} u_{i} (0) = 0 u_{i}^{'} (0) = 0 u_{i}^{'} (1) = \sum_{j = 1}^{p} β_{j, i} u_{i}^{'} (η_{j, i}) + \int_{0}^{1} h_{i} (u_{1} (s), \dots, u_{n} (s)) d s \end{matrix}$ where $β_{j, i} > 0$ , $0 < η_{1, i} < \dots < η_{p, i} < \frac{1}{2}$ , $f_{i} : [0, 1] \times ℝ^{n} \times ℝ^{n} \to ℝ$ and $h_{i} : [0, 1] \times ℝ^{n} \to ℝ$ are continuous functions for all $i \in {1, \dots, n}$ and $j \in {1, \dots, p}$ . Using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.

Region of variability for spiral-like functions with respect to a boundary point

S. Ponnusamy, A. Vasudevarao, M. Vuorinen (2009)

Colloquium Mathematicae

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For μ ∈ ℂ such that Re μ > 0 let $ℱ_{μ}$ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and $R e (2 π / μ z f^{'} (z) / f (z) + (1 + z) / (1 - z)) > 0$ in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class $ℱ_{μ} (λ) = f \in ℱ_{μ} : f^{'} (0) = (μ / π) (λ - 1) a n d f^{''} (0) = (μ / π) (a (1 - | λ | ²) + (μ / π) (λ - 1) ² - (1 - λ ²))$ . In the final section we graphically illustrate the region of variability for several sets of parameters.