A note on Mathieu functions

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

Asymptotic behavior in time of the solutions of three nonlinear partial differential equations

This paper is an extension of a recent work done by the author [4] and here the sums of alternating series of odd powers (up to fifteen) of the reciprocals of odd positive integers are computed. Following this method, the sum of the series of any higher power could be calculated. In the process of computing these sums, the sums of the series of even powers of reciprocals of odd positive integers have been reestablished and enabled the author to compute the values of Riemann’s zeta function for even positive integers.

Commuting graph for a finite group G, denoted by ΓG, with its set of vertices G∖Z(G), where Z(G) is the center of G, is a graph with vp,vq∈G∖Z(G), vp≠vq, are adjacent whenever vpvq=vqvp. In recent years, there has been significant research into the energy of graphs, particularly focusing on matrices associated with the degree of vertices. Therefore, motivated by that, our study elaborates on the energy of ΓG for dihedral groups of order 2n, D2n, concerning some graph matrices related to the degree of elements of D2n∖Z(D2n) and examine the correlation between those energies. The matrices involved are known as geometric-arithmetic, symmetric division deg, degree exponent, inverse sum indeg and Sombor matrices. Based on these five matrices, it is found that the lowest graph energy is the geometric-arithmetic energy of ΓG whilst the highest is the degree exponent energy. Furthermore, the geometric-arithmetic, symmetric division deg, and degree exponent energies are always positive even integers. In contrast, the inverse sum indeg energy is a positive integer that can be either even or odd. Meanwhile, the Sombor energy is never an odd integer.

Let a be a positive integer with $a>1$ , and let $(x, y, n)$ be a positive integer solution of the equation $x^{2}+a^{2}=y^{n}$ , $\gcd(x, y)=1$ , $n>2$ . Using Baker’s method, we prove that, for any positive number ϵ, if n is an odd integer with $n>C(\epsilon)$ , where $C(\epsilon)$ is an effectively computable constant depending only on ϵ, then $n<(2+\epsilon)(\log a)/\log y$ . Owing to the obvious fact that every solution $(x, y, n)$ of the equation satisfies $n>2(\log a)/\log y$ , the above upper bound is optimal.

In the present paper I obtain an extension of the so-called Eisenstein theorem which is proved by means of RiemannStieltjes integration. THEOREM. Let n be a positive integer >2. Let P1.P2, p.. ,Pn be odd integers and relatively prime in pairs. Then n (pi-1)/2 n P 1 n i=1 x=1 k=l;k*i L Pi k=1 PROOF. The method of proof is Riemann-Stieltjes integration. Consider the integrals r1/2 1 = [P1X] [p2x] ... [Pn-X] d[pnx], 1/2 2 = [PnX] d[p1x] [P2x] . [Pn-lX] First we observe that the greatest integer functions [plx], [p2x],*** [pAx] have no discontinuities in common on the interval O<x l in view of the condition on the integers Pl, P2, , Pn. Second, the discontinuities of [p x], i=1, 2, , n, on the interval O<x_ are at x=klpi, k=1, 2, * , *21(pi-1), and the jump of [pix] at each of these discontinuities is equal to 1. Hence the value of Y1 is 2 [plx] [p2x] * [pn_x], k = 1, 2, . . . 21(pn1) x=k1p, or (Pn-l)/2 rPlXrP2Xl -Pn 1X Consider the second integral. The discontinuities of [p1x] [p2x] . . [pn_IX] are at x=klpi, k=1,2, ,21(pi-l), i=1,2, ,n-1. Since there are no common discontinuities and the jumps of [pix] at the discontinuities k/pi are equal to 1, the jump of [plx] [p2x] ... [Pn-IX] at each Received by the editors May 9, 1973. AMS (MOS) subject classifications (1970). Primary 1OA20, 1OA25. ? American Mathematical Society 1974 69 This content downloaded from 157.55.39.243 on Wed, 05 Oct 2016 04:52:27 UTC All use subject to http://about.jstor.org/terms

Noether's problem for some semidirect products

Erdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2x + by = cz is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2r − 1, r + 2,2, 2), where r is a positive integer with r ≥ 2.

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

Suppose {Xn n ≥ 1} is a stationary sequence satisfying the D and D1 mixing conditions given by Adler [1]. Suppose further that g and h are functions on the positive integers such that h is positive, periodic with integer period p ≥ 1 and g satisfies a certain growth rate condition. By first exhibiting the weak convergence of a sequence of point processes related to , we derive the asymptotic distribution of . When {Xn} is in the maximal domain of attraction of A(x) = exp(-e -xx), {Mn} is attracted to (A(x))q/p where . If {Xn} is in the maximal domain of attraction of G(x) = exp(-x -α) then {Mn} is attracted to (G(x)) θ where . These asymptotic results lead to a new method for estimating the level of the k-year return period, a quantity of interest in many climatological studies. An analysis of the daily rainfall totals in Homestead, Florida from 1939-1979 is provided, illustrating this estimation procedure.

Let IN be the set of positive integers, $${\cal B}$$ = {b1 < ⋅s < bh}⊂IN, N∊IN and N≥ bh. $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sarkozy) such that $${\cal A}$$ {1,..., N} = $${\cal B}$$ and p( $${\cal A}$$ ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( $${\cal A}$$ ,n) denotes the number of partitions of n with parts in $${\cal A}$$ . Let us denote by σ ( $${\cal A}$$ ,n) the sum of the divisors of n belonging to $${\cal A}$$ . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( $${\cal A}$$ ,2k n))n≥ 1mod 2k+1 is periodic with an odd period qk. In this paper, we will characterize for any fixed odd positive integer q, the sets $${\cal B}$$ and the integers N such that q0 = q, and those for which qk = q for all k≥ 0. Moreover, a set $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is constructed with the property that its period, i.e. the period of (σ( $${\cal A}$$ ,n))n≥ 1mod 2, is 217, and for which the counting function is asymptotically equal to that of $${\cal A}$$ 0({1,2,3,4,5},5) which is a set of period 31.

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L

New constructions of strongly regular Cayley graphs on abelian non p-groups