A Lucas–Lehmer approach to generalised Lebesgue–Ramanujan–Nagell equations

We describe a computationally efficient approach to resolving equations of the form $$C_1x^2 + C_2 = y^n$$ C 1 x 2 + C 2 = y n in coprime integers, for fixed values of $$C_1$$ C 1 , $$C_2$$ C 2 subject to further conditions. We make use of a factorisation argument and the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier.

Zero Set Structure of Real Analytic Beltrami Fields

In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in $${\mathbb {R}}^3$$ R 3 , there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue $$+1$$ + 1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

BPS invariants for 3-manifolds at rational level K

We consider the Witten-Reshetikhin-Turaev invariants or Chern-Simons partition functions at or around roots of unity $$ q={e}^{\frac{2\pi i}{K}} $$ q = e 2 πi K with a rational level K = $$ \frac{r}{s} $$ r s where r and s are coprime integers. From the exact expression for the G = SU(2) Witten-Reshetikhin-Turaev invariants of the Seifert manifolds at a rational level obtained by Lawrence and Rozansky, we provide an expected form of the structure of the Witten-Reshetikhin-Turaev invariants in terms of the homological blocks at a rational level. Also, we discuss the asymptotic expansion of knot invariants around roots of unity where we take a limit different from the limit in the standard volume conjecture.

The number of maximum primitive sets of integers

A set of integers is primitive if it does not contain an element dividing another. Let f(n) denote the number of maximum-size primitive subsets of {1,…,2n}. We prove that the limit α = lim n→∞ f(n)1/n exists. Furthermore, we present an algorithm approximating α with (1 + ε) multiplicative error in N(ε) steps, showing in particular that α ≈ 1.318. Our algorithm can be adapted to estimate the number of all primitive sets in {1,…,n} as well. We address another related problem of Cameron and Erdős. They showed that the number of sets containing pairwise coprime integers in {1,…n} is between ${2^{\pi (n)}} \cdot {e^{(1/2 + o(1))\sqrt n }}$ and ${2^{\pi (n)}} \cdot {e^{(2 + o(1))\sqrt n }}$ . We show that neither of these bounds is tight: there are in fact ${2^{\pi (n)}} \cdot {e^{(1 + o(1))\sqrt n }}$ such sets.

Necessary condition for the existence of a simple closed geodesic on a regular tetrahedron in the spherical space

In the spherical space the curvature of the tetrahedron’s faces equals 1, and the curvature of the whole tetrahedron is concentrated into its vertices and faces. The intrinsic geometry of this tetrahedron depends on the value α of faces angle, where π/3 < α ⩽ 2π/3. The simple (without points of self-intersection) closed geodesic has the type (p,q) on a tetrahedron, if this geodesic has p points on each of two opposite edges of the tetrahedron, q points on each of another two opposite edges, and (p+q) points on each edges of the third pair of opposite one. For any coprime integers (p,q), we present the number αp, q (π/3 < αp, q < 2π/3) such that, on a regular tetrahedron in the spherical space with the faces angle of value α > αp, q, there is no simple closed geodesic of type (p,q)

The largest size of an (s,s + 1)-core partition with parts of the same parity

Finding the largest size of a partition under certain restrictions has been an interesting subject to study. For example, it is proved by Olsson and Stanton that for two coprime integers [Formula: see text] and [Formula: see text], the largest size of an [Formula: see text]-core partition is [Formula: see text]. Xiong found a formula for the largest size of a [Formula: see text]-core partitions with distinct parts. In this paper, we find an explicit formula for the largest size of an [Formula: see text]-core partition such that all parts are odd (or even).

On a class of critical elliptic systems in ℝ4

In the present paper, we consider the following classes of elliptic systems with Sobolev critical growth: $$\begin{array}{} \displaystyle \begin{cases} -{\it\Delta} u+\lambda_1u=\mu_1 u^3+\beta uv^2+\frac{2q}{p} y u^{\frac{2q}{p}-1}v^2\quad &\hbox{in}\;{\it\Omega}, \\ -{\it\Delta} v+\lambda_2v=\mu_2 v^3+\beta u^2v+2 y u^{\frac{2q}{p}}v\quad&\hbox{in}\;{\it\Omega}, \\ u,v \gt 0&\hbox{in}\;{\it\Omega}, \\ u,v=0&\hbox{on}\;\partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝ4 is a smooth bounded domain with smooth boundary ∂Ω; p, q are positive coprime integers with 1 < $\begin{array}{} \displaystyle \frac{2q}{p} \end{array}$ < 2; μi > 0 and λi ∈ ℝ are fixed constants, i = 1, 2; β > 0, y > 0 are two parameters. We prove a nonexistence result and the existence of the ground state solution to the above system under proper assumptions on the parameters. It seems that this system has not been explored directly before.

The generalized Fermat equation with exponents 2, 3,

We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$, to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$. We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti-symplectically isomorphic $p$-torsion modules. Using these criteria we produce the minimal list of twists of $X(p)$ that have to be considered, based on local information at 2 and 3; this list depends on $p\hspace{0.2em}{\rm mod}\hspace{0.2em}24$. We solve the equation completely when $p=11$, which previously was the smallest unresolved $p$. One new ingredient is the use of the ‘Selmer group Chabauty’ method introduced by the third author, applied in an elliptic curve Chabauty context, to determine relevant points on $X_{0}(11)$ defined over certain number fields of degree 12. This result is conditional on the generalized Riemann hypothesis, which is needed to show correctness of the computation of the class groups of five specific number fields of degree 36. We also give some partial results for the case $p=13$. The source code for the various computations is supplied as supplementary material with the online version of this article.

INFINITUDE OF PRIME p such that ap+b is prime where a, b are coprime integers

In 1904, Dickson [6] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [3]generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to themultivariable case. In 2006, Green and Tao [9] considered Dickson’sconjecture in the multivariable case and gave directly a generalizedHardy-Littlewood estimation. But, the precise Dickson’s conjecture inthe multivariable case does not seem to have been formulated. In thispaper, based on the idea in [8] we introduce an interesting class of prime numbers to solve the dickson conjecture Although this article does not solve the dickson conjecture but it solves a problem that is similar to the Dickson conjecture. the problem is stated as follows being given two coprime integers a, b there is an infinity of prime numbers p such that ap+b is prime. This type of prime numbers we call it Bado-Tiemoko prime numbers .We intend to generalize this result but for the moment we speculate that given a family $(a_{i},b_{i})_{1\leq i\leq k}$ such that $\gcd(a_{i},b_{i})=1,\forall 1\leq i\leq k$ there is an infinity of prime numbers $p$ such that $a_{i}p+b_{i}$ is prime for $\forall 1\leq i\leq k$ Let $q_{a}$ be the smallest prime number dividing $a$ and $\omega(q_{a})$ its order by arranging the prime numbers in ascending order.Let $\beta(n)$ the number of Bado-Tiemoko prime less than $n$ and $$\eta_{s,i}=\frac{(p_{i}-1)^{\delta_{p_{i}}(a)}\prod_{k=1}^{s}(p_{i_{k}}-1)^{\delta_{p_{i_{k}}(a)}}}{\phi(a)\prod_{k=1}^{s}(p_{i_{k}}-1)(p_{i}-1)\prod_{k=1}^{s}p_{i_{k}}^{\delta_{p_{i_{k}}(a)}}p_{i}^{\delta_{p_{i}}(a)}}$$ $$\mu(r)=\frac{1}{\phi(a)}[\sum_{s=1}^{\omega(q_{a})}(-1)^{s-1}\sum_{1\leq i_{1}

Predicting Maximal Gaps in Sets of Primes

Let q > r ≥ 1 be coprime integers. Let P c = P c ( q , r , H ) be an increasing sequence of primes p satisfying two conditions: (i) p ≡ r (mod q) and (ii) p starts a prime k-tuple with a given pattern H. Let π c ( x ) be the number of primes in P c not exceeding x. We heuristically derive formulas predicting the growth trend of the maximal gap G c ( x ) = max p ′ ≤ x ( p ′ − p ) between successive primes p , p ′ ∈ P c. Extensive computations for primes up to 10 14 show that a simple trend formula G c ( x ) ∼ x π c ( x ) · ( log π c ( x ) + O k ( 1 ) ) works well for maximal gaps between initial primes of k-tuples with k ≥ 2 (e.g., twin primes, prime triplets, etc.) in residue class r (mod q). For k = 1, however, a more sophisticated formula G c ( x ) ∼ x π c ( x ) · log π c 2 ( x ) x + O ( log q ) gives a better prediction of maximal gap sizes. The latter includes the important special case of maximal gaps in the sequence of all primes (k = 1 , q = 2 , r = 1). The distribution of appropriately rescaled maximal gaps G c ( x ) is close to the Gumbel extreme value distribution. Computations suggest that almost all maximal gaps satisfy a generalized strong form of Cramér’s conjecture. We also conjecture that the number of maximal gaps between primes in P c below x is O k ( log x ).