It is well known that, if p is prime, the multiplicative group (ℤ/pℤ)* of reduced residues is cyclic. A generator is called a primitive root; there are φ(p − 1) of them, where φ is Euler's function. Thus, (φ(p − 1)/(p−1) is the proportion of primitive roots modulo p in (ℤ/pℤ)*. Elliott has proved that φp − 1)/(p − 1) has a limiting distribution function [2], in the sense that

exists for all real numbers u.

It is shown that the regular n-dimensional simplex has a non-convex cross-section body for all n ≥ 4.

Let φ(n) be the Euler function (i.e., φ(n) denotes the number of integers less than n which are relatively prime to n), and define

These functions were extensively studied by several mathematicians. One of the problems investigated concerns their sign changes. We say that a function fx) has a sign change at x = x0 if f(x0 −) f(x0 +) < 0, and f(x) has a sign change on the integer n if (n)f(n+1) < 0. The numbers of sign changes and sign changes on integers of f(x) in the interval [1, T] are denoted by Xf(T) and Nf(T), respectively.

It is shown that the cross-section body of a convex body K ⊂ ℝ3, that is the symmetric body which has for radial function in the direction u the maximal volume of a section of K by an hyperplane orthogonal to u, is a convex body in ℝ3.

It is shown that every compact convex set K which is centrally symmetric and has a non-empty interior admits a packing of Euclidean 3-space with density at least 0.46421 … The best such bound previously known is 0.30051 … due to the theorem of Minkowski-Hlawka. It is probable that there is such a lower bound which is significantly greater than the one shown in this note, since there is a packing of congruent spheres which has density

Assuming the abc-conjecture, it is shown that there are only finitely many powerful binomial coefficients with 3≤k≤n/2 in fact, if q2 divides , then . Unconditionally, it is shown that there are N1/2+σ(1) powerful binomial coefficients in the top N rows of Pascal's Triangle.

We consider the spectral function

associated with the Sturm-Liouville equation

in situations where q(x) →−∞ as x → ∞ and (1.1) is in the Weyl limit-point case at ∞. As usual, q is real-valued and locally integrable in [0, ∞], and our particular concern is where q(x) has the form

where c (>0) is a parameter, s and p are non-negative on [0, ∞], p(x) → ∞ and p(x) = 0{s(x) } as x → ∞. As the boundary condition at x = 0, we take the Dirichlet condition y(0) = 0 for convenience: we can equally take the Neumann condition y′ (0) = 0 or generally a1y(0) + a2y′ (0) = 0 with real a1 and a2.

A long-standing conjecture claims that the Diophantine equation

has finitely many solutions, and, maybe, only those given by

In 1984 Heath-Brown [2] proved the conjecture of Erdős and Mirsky [1], and obtained the following result.

Theorem. There are infinitely many integers n for which d(n+ 1)= d(n). Indeed, for large x, the number of such n≤x is at least of orderx ln-7x.

It is well known that boundary value problems for hyperbolic equations are in general “not well posed” problems. This paper is concerned with the uniqueness of solutions to boundary value problems for the hyperbolic equation uxx − Qu = utt. Here Q is a function of the variable x alone, and satisfies the following conditions:

(a) Q:[0, ∞) → ℝ;

(b) Q is Lebesgue integrable on any compact subinterval of [0, ∞);

(c) Q(x)→ ∞ as x → ∞.

We prove a concentration inequality for δ-concave measures over ℝn. Using this result, we study the moments of order q of a norm with respect to a δ-concave measure over ℝn. We obtain a lower bound for q∈ ]−1, 0] and an upper bound for q∈ ]0,+ ∞[ in terms of the measure of the unit ball associated to the norm. This allows us to give Kahane-Khinchine type inequalities for negative exponent.

It is shown that an integral domain R has the property that every pure submodule of a finite direct sum of ideals of R is a summand if and only if R is an h-local Prüfer domain; equivalently, (J + K:I) = (J:I) + (K:I) for all ideals I, J and K of R. These results are extended to submodules of the quotient field of an integral domain.

Let be a class of structures for a first-order language and let n be a positive integer. A structure A for the language is said to be n-residually if, given elements a1, … an and b1, … bn of A with ai≠bi for 1 ≤i≤n, there exist B ∈ and an epimorphism φ:A → B such that φ(ai≠φ(bi for 1 ≤i ≤ n. We abbreviate 1 -residually to residually , and A is said to be fully residually if it is n-residually for all n ≤ 1. We shall only be using two cases. One is where the language is the first-order language of groups, {·−1,1}, and A and all members of are groups. In this case we may take all the bi in the definition to be equal to 1. However, we are mainly interested in the first-order language of rings, {+,−·,0,1}. Again if A and all members of are rings, we may take all bi to be zero in the definition.

The existence of global solutions to the discrete coagulation equations is investigated for a class of coagulation rates of the form ai, j = rirj + αi, j with αi, j≤Krirj. In particular, global solutions are shown to exist when the sequence (ri) increases linearly or superlinearly with respect to i. In this case also, the failure of density conservation (indicating the occurrence of the gelation phenomenon) is studied.

Let p(n) be the usual partition function. Let l be an odd prime, and let r (mod t) be any arithmetic progression. If there exists an integer n ≡ r (mod t) such that p(n) ≢ 0 (mod l), then, for large X,

This paper treats the reducibility of the quasiperiodic linear differential equations

where A is a constant matrix with multiple eigenvalues, Q(t) is a quasiperiodic matrix with respect to time t, and ε is a small perturbation parameter. Under some non-resonant conditions, rapidly convergent methods prove that, for most sufficiently small ε, the differential equations are reducible to a constant coefficient differential equation by means of a quasiperiodic change of variables with the same frequencies as Q(t).

In the paper [5] bounds are found for the sum,

for a suitable Dirichlet character χ mod r, and real functionf(x). The proofs in that paper use Bombieri and Iwaniec's method [1], one formulation of which has as part of its first step the estimation of S in terms of a sum of many shorter sums of the form,

where e(x) = exp (2πix), mi∈ [M, 2M], and each mi, lies in its own interval, of length N ≥ M/4, that is disjoint from those of the others. This paper addresses a problem springing from above: to bound the numbers of ‘similar’ pairs, Si+, Si+, satisfying both

and

where ‖x‖ = min{|x − n|: n ∈ ℤ}. Lemma 5.2.1 of [3] (partial summation) shows that each sum in a similar pair is a bounded multiple of the other.

Consider the equation

where , . The inversion problem for (1) is called regular in Lp if, uniformly in p∈[1, ∞] for any f(x)∈ Lp(R), equation (1) has a unique solution y(x)∈ Lp(R) of the form

with . Here G(x, t) is the Green function corresponding to (1) and c is an absolute constant. For a given s∈[l, ∞], necessary and sufficient conditions are obtained for assertions (2) and (3) to hold simultaneously:

(2) the inversion problem for (1) is regular in Lp;

(3) for any f(x)∈LS(R).

A universal algebra is called congruence compact if every family of congruence classes with the finite intersection property has a non-empty intersection. This paper determines the structure of all right congruence compact monoids S for which Green's relations ℐ and ℋ coincide. The results are thus sufficiently general to describe, in particular, all congruence compact commutative monoids and all right congruence compact Clifford inverse monoids.