Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)^r = λ(n)^s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.     

Prime analogues of the Erd?s-Kac theorem for elliptic curves

Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. We denote by ω(#E(Fp)), the number of distinct prime divisors of #E(Fp). We prove that the quantity (assuming the GRH if E is non-CM)

     

A problem of Chowla revisited

In 1964, S. Chowla asked if there is a non-zero integer-valued function f with prime period p such that f(p)=0 and

     

Polynomial Cunningham chains

A sequence of prime numbers p₁,p₂,p₃,…, such that pi=2pi−1+? for all i, is called a Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the smallest positive integer such that 2pk+? is composite, then we say the chain has length k. It is conjectured that there are infinitely many Cunningham chains of length k for every positive integer k. A sequence of polynomials f₁(x),f₂(x),… in Z[x], such that f₁(x) has positive leading coefficient, each fi(x) is irreducible in Q[x] and fi(x)=xfi−1(x)+? for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the least positive integer such that fk+1(x) is reducible in Q[x], then we say the chain has length k. In this article, for polynomial Cunningham chains of both kinds, we prove that there are infinitely many chains of length k and, unlike the situation in the integers, that there are infinitely many chains of infinite length, by explicitly giving infinitely many polynomials f₁(x), such that fk+1(x) is the only term in the sequence that is reducible.     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

On the prime power factorization of n!, II

Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)|n and pep(n)+1?n. The following results are proved: (1) For any positive integer m, any prime p and any ε∈Zm, there are infinitely many positive integers n such that ; (2) For any positive integer m, there exists a constant D(m) such that if ε,δ∈Zm and p, q are two distinct primes with max{p,q}?D(m), then there exist infinitely many positive integers n such that , . Finally we pose four open problems.     

Computation of the Iwasawa invariants of certain real abelian fields

Let p be a prime number and k a finite extension of . It is conjectured that the Iwasawa invariants λ_p(k) and μ_p(k) vanish for all p and totally real number fields k. Some methods to verify the conjecture for each real abelian field k are known, in which cyclotomic units and a set of auxiliary prime numbers are used. We give an effective method, based on the previous one, to compute the exact value of the other Iwasawa invariant ν_p(k) by using Gauss sums and another set of auxiliary prime numbers. As numerical examples, we compute the Iwasawa invariants associated to in the range 1<f<200 and 5?p<10000.     

A Bombieri-Vinogradov type exponential sum result with applications

We prove a Bombieri-Vinogradov type result for linear exponential sums over primes. Then we apply it to show that, for any irrational α and some θ>0, there are infinitely many primes p such that p+2 has at most two prime factors and ‖αp+β‖<p−θ.     

Exact number of positive solutions for semipositone problems

This paper is concerned with the exact number of positive solutions for the boundary value problem ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which both f(u) and g(u)=(p−1)f(u)−uf^′(u) change sign exactly once from negative to positive on (0,∞).     

A complete determination of Rabinowitsch polynomials

Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x²+x−m. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x₀,x₀+1,…,x₀+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).     

Exact multiplicity results for a class of two-point boundary value problems

This paper is concerned with the exact number of positive solutions for boundary value problems ^′(|y^′|^p−2y^′)+λf(y)=0 and y(−1)=y(1)=0, where p>1 and λ>0 is a positive parameter. We consider the case in which the nonlinearity f is positive on (0,∞) and (p−1)f(u)−uf^′(u) changes sign from negative to positive.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

Torsion points on some abelian surfaces of CM-type

Let A be a two-dimensional abelian variety of CM-type defined over Q, which is not simple over C. Let p be a prime number. We show that torsion points of A(Q) of prime order p are possible only for p≦7.     

Uniqueness results for the one-dimensional p-Laplacian

This paper is concerned with positive solutions of the boundary value problem ^′(|y^′|^p−2y^′)+f(y)=0, y(−b)=0=y(b) where p>1, b is a positive parameter. Assume that f is continuous on (0,+∞), changes sign from nonpositive to positive, and f(y)/yp−1 is nondecreasing in the interval of f>0. The uniqueness results are proved using a time-mapping analysis.     

On the enumeration of finite Abelian and solvable groups

Let a(n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a(n). More generally, we get short interval version of results of Ivi? on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdös and Szekeres on the summatory function of a(n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups.     

On the prime power factorization of n!

For a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For two fixed integers m?2 and r with 0?r?m−1, let A(x;m,q,r) denote the numbers of positive integers n?x for which . In this paper we shall prove a sharp asymptotic formula of A(x;m,q,r).     

On linear equations with prime variables of special type

Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. It is proved that for every sufficiently large odd integer , the equation p₁+p₂+p₃=n is solvable in prime variables p₁,p₂,p₃ such that p₁+2=P₂, , and for almost all sufficiently large even integer , the equation p₁+p₂=n is solvable in prime variables p₁,p₂ such that p₁+2=P₂.     

Short effective intervals containing primes

We prove that every interval ]x(1−Δ⁻¹),x] contains a prime number with Δ=28314000 and provided x?10726905041. The proof combines analytical, sieve and algorithmical methods.     

Reciprocal relations between cyclotomic fields

We describe a reciprocity relation between the prime ideal factorization, and related properties, of certain cyclotomic integers of the type ?n(c−ζm) in the cyclotomic field of the m-th roots of unity and that of the symmetrical elements ?m(c−ζn) in the cyclotomic field of the n-th roots. Here m and n are two positive integers, ?n is the n-th cyclotomic polynomial, ζm a primitive m-th root of unity, and c a rational integer. In particular, one of these integers is a prime element in one cyclotomic field if and only if its symmetrical counterpart is prime in the other cyclotomic field. More properties are also established for the special class of pairs of cyclotomic integers q(1−ζp)−1 and p(1−ζq)−1, where p and q are prime numbers.