On the existence of reflecting n-queens configurations

Let us denote by { m, n } the number of distinct pairs of integers, r and s, such that* (r, s) = 1, 1 ≤ r ^ m, 1 ≤ s ^ n.

Let m>1 and n>1 be any pair of integers. In this paper we prove that if H is between the numbers \cot(\frac{\pi}{m}) and b_{m,n}=\frac{(m^2-2)\sqrt{n-1}}{n\sqrt{m^2-1}}, then, there exists a non isoparametric, compact embedded hypersurface in S^{n+1} with constant mean curvature H that admits the group O(n)x Z_m in their group of isometries, here O(n) is the set of n x n orthogonal matrices and Z_m are the integers mod m. When m=2 and H is close to the boundary value 0, the hypersurfaces look like two very close n-dimensional spheres with two catenoid necks attached, similar to constructions made by Kapouleas. When m>2 and H is close to \cot(\frac{\pi}{m}), the hypersurfaces look like a necklets made out of m spheres with (m+1) catenoid necks attached, similar to constructions made by Butscher and Pacard. In general, when H is close to b_{m,n} the hypersurface is close to an isoparametric hypersurface with the same mean curvature. As a consequence of the expression of these bounds for H, we have that every H different from 0,\pm\frac{1}{\sqrt{3}} can be realized as the mean curvature of a non isoparametric CMC surface in S^3. For hyperbolic spaces we prove that every non negative H can be realized as the mean curvature of an embedded CMC hypersurface in H^{n+1}, moreover we prove that when H>1 this hypersurface admits the group O(n)\times Z in its group of isometries. Here Z are the integer numbers. As a corollary of the properties proven for these hypersurfaces, for any n> 5, we construct non isoparametric compact minimal hypersurfaces in S^{n+1} which cone in R^{n+2} is stable. Also, we will prove that the stability index of every non isoparametric minimal hypersurface with two principal curvatures in S^{n+1} is greater than 2n+5.

There are two methods of constructing mathematical theories: genetic and axiomatic. The first of them implies the construction of a theory through successive generalizations, on the basis of simple concepts established earlier. For instance, one can construct in this way the theory of real numbers proceeding from the basic unit: by applying a certain natural process (that of counting) one obtains from it the series of positive integers 1, 2, 3, 4, 5,…; introduces the so called arithmetic operations for them; then, by making use of these operations, introduces the negative numbers and zero; then fractions as pairs of integers; and, finally, defines the real number as a section or fundamental sequence.

This paper introduces a novel Recursive Partitioning Framework that builds upon additive number theory, with specific application to Lemoine's Conjecture, which asserts that every odd integer greater than 5 can be expressed as the sum of a prime and a semiprime. Inspired by recent developments in algorithmic formulations of Goldbachtype conjectures, we adapt the framework proposed by Sankei et al. (2023), originally used to partition even integers via expressions of the form; 𝐸 = (𝑃1 + 𝑃2 )+ (𝑃2 −𝑃1 ) 𝑛 with 𝑃1 , 𝑃2 ∈ ℙ, 𝑃2 > 𝑃1 , and 𝑛 ∈ ℕ, to develop a systematic method for generating and verifying odd number partitions, tested for all odd numbers up to 106 . Our method leverages structured arithmetic sets and recursions over integer pairs (𝑒, 𝑢), where 𝑒 ∈ 2ℤ and 𝑢 ∈ 2ℤ + 1, to explore partitions of an odd integer 𝑂 = 𝑝 + 𝑠, where 𝑝 is an odd prime and 𝑠 is a semiprime. A recursive algorithm is proposed that decomposes residual values resulting from candidate partitions into products of two primes. The method reduces computational complexity compared to brute-force approaches by exploiting arithmetic patterns and interval narrowing based on parity constraints. Empirical validation confirms the algorithm consistently finds valid Lemoine decompositions for all tested odd integers 𝑂 > 5. Furthermore, we define a Lemoine pair function 𝑓(𝑂), which asymptotically satisfies 𝑓(𝑂) ≳ 𝑐 ⋅ 𝑂𝑙𝑜𝑔 𝑙𝑜𝑔 𝑂 𝑙𝑜𝑔2 𝑂 , suggesting the unbounded growth of valid partitions with increasing 𝑂. This offers a probabilistic foundation for the conjecture's global validity. The recursive partitioning framework not only unifies prime-semiprime decompositions with structured partition theory, but also opens new directions in analytic number theory and cryptography by enabling systematic methods for prime generation relevant to cryptographic protocols

The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R.P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallée modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm; however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D.E. Knuth in The Art of Computer Programming.

The notion of variable, or unknown, appeared in the works of the Greek mathematician Diophantus, who lived (probably) during the third century a.d. He was particularly interested in the following question: does a given polynomial equation with integral (or rational) coefficients have a solution in integers (or in rational numbers)? Among the most classical examples is the equation x 2 + y 2 = z 2 , whose integral solutions give us the lengths of the sides of Pythagorean triangles. At that time (and, most probably, even since a few centuries before that time), all these solutions were perfectly known. Nowadays, we call Diophantine equation any polynomial equation with integer coefficients and whose unknowns are supposed to be rational integers. This definition is often extended to any type of equations involving integers and where the unknown are also integers. An emblematic example is Fermat's equation x n + y n = z n , where x, y, z and n > 3 are unknown positive integers. We often use the terminology “exponential Diophantine equation” when one or more exponents are unknown. The natural question is the following: an equation being given, determine the complete set of its integral solutions. Sometimes, this is quite easy, in particular when one can use congruences modulo a suitable integer. Let us for example consider the equation 3 m - 2 n = 1, which was solved by Levi ben Gershon (1288-1344), answering a question of the French composer Philippe de Vitry. Assume that there are integers m, n with n > 2 and 3m - 2n = 1. Then, 4 divides 3m - 1, whence m must be even. Writing m = 2k we obtain (3 k - 1) (3 k + 1) = 2n, which implies that both 3 k - 1 and 3 k + 1 are powers of 2. But the only powers of 2 which differ by 2 are 2 and 4. Hence k = 1 and we have proved that 3 2 - 2 3 = 1 is the only solution to 3 m - 2 n = 1 with n > 2. However, in most of the cases, to determine the complete set of integral solutions of a Diophantine equation remains an unsolved problem, and often it is even very difficult to prove whether this set is finite or not. When it is infinite, the next step is to give a complete description of all the integral solutions of the equation. For instance, the positive solutions of the equation 5x 2 - y 2 = ± 4 are precisely given by the integer pairs (Fn, Ln), where (Fn)n1 and (Ln)n1 are the Fibonacci and the Lucas sequences defined by F1 = F2 = 1, L1 = 1, L2 = 3, and satisfying Fn+2 = Fn+1 + Fn and Ln+2 = Ln+1 + Ln, for n > 1.

For an integer $k\ge 2$, a tuple of $k$ positive integers $(M_{i})_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \begin{equation*}\sigma (M_{1})=\cdots =\sigma (M_{k})=M_{1}+\cdots +M_{k}\end{equation*} holds. This is a generalization of amicable pairs. An amicable pair is a pair of distinct positive integers each of which is the sum of the proper divisors of the other. Gmelin (Über vollkommene und befreundete Zahlen (1917) Heidelberg University) conjectured that there is no relatively prime amicable pairs and Artjuhov (Acta Arith. 27 (1975) 281–291) and Borho (Math. Ann. 209 (1974) 183–193) proved that for any fixed positive integer $K$, there are only finitely many relatively prime amicable pairs $(M,N)$ with $\omega (MN)=K$. Recently, Pollack (Mosc. J. Comb. Number Theory 5 (2015), 36–51) obtained an upper bound \begin{equation*}MN<(2K)^{2^{K^{2}}}\end{equation*} for such amicable pairs. In this paper, we improve this upper bound to \begin{equation*}MN<\frac{\pi^{2}}{6}2^{4^{K}-2\cdot 2^{K}}\end{equation*} and generalize this bound to some class of general amicable tuples.

Reciprocal relations between cyclotomic fields

Heron triangles with two fixed sides

Let ϕ ( n ) \phi (n) denote Euler’s totient function, i.e., the number of positive integers &gt; n &gt;n and prime to n n . We study pairs of positive integers ( a 0 , a 1 ) (a_{0},a_{1}) with a 0 ≤ a 1 a_{0}\le a_{1} such that ϕ ( a 0 ) = ϕ ( a 1 ) = ( a 0 + a 1 ) / k \phi (a_{0})=\phi (a_{1})=(a_{0}+a_{1})/k for some integer k ≥ 1 k\ge 1 . We call these numbers ϕ \phi –amicable pairs with multiplier k k , analogously to Carmichael’s multiply amicable pairs for the σ \sigma –function (which sums all the divisors of n n ). We have computed all the ϕ \phi –amicable pairs with larger member ≤ 10 9 \le 10^{9} and found 812 812 pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other ϕ \phi –amicable pairs can be associated. Among these 812 812 pairs there are 499 499 so-called primitive ϕ \phi –amicable pairs. We present a table of the 58 58 primitive ϕ \phi –amicable pairs for which the larger member does not exceed 10 6 10^{6} . Next, ϕ \phi –amicable pairs with a given prime structure are studied. It is proved that a relatively prime ϕ \phi –amicable pair has at least twelve distinct prime factors and that, with the exception of the pair ( 4 , 6 ) (4,6) , if one member of a ϕ \phi –amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive ϕ \phi –amicable pairs with larger member &gt; 10 9 &gt;10^{9} , the largest pair consisting of two 46-digit numbers.

Universally bad integers and the 2-adics

Universally bad integers and the 2-adics

The standard binary reflected Gray code produces a permutation of the sequence of integers (0, 1,…, n − 1), where n is a power of 2, such that the binary representation of each integer in the permuted sequence differs from the binary representation of the preceding integer in exactly one bit. In an earlier paper, we presented two methods to compute binary dense Gray codes, which extend the possible values of n to the set of all positive integers while preserving both the Gray-code property — only one bit changes between each pair of consecutive integers — and the denseness property — the sequence contains exactly the n integers 0 to n − 1. This paper generalizes our method for binary dense Gray codes to arbitrary radices that may be either a single fixed radix for all digits or mixed radices, so that each digit may have a different radix. That is, we show how to produce a permutation of (0, 1, …, n−1) represented in any set of radices, such that the representation of each number differs from the representation of the preceding number in exactly one digit, and the values of these digits differ by exactly 1. We provide a simple formula for this permuation, which we can use to quickly compute a Hamiltonian path for a dynamic array of n nodes, where the nodes are added and deleted in order along the k dimensions of a grid network.

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm x or less as x goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdös.

On the degrees of the vertices of a directed graph