Michael Filaseta

On gaps between squarefree numbers II

A squarefree number is a positive integer not divisible by the square of an integer > 1. We investigate here the problem of finding small h = h(x) such that for x sufficiently large, there is a squarefree number in the interval (x,x + h]. This problem was originally investigated by Fogels [3]; he showed that for every ∈ > 0, h = x 2/5+∈ is admissible. Later Roth [9] reported elementary arguments of Davenport and Estermann showing respectively that one can take h ≫ x 1/3 and h ≫ x 1/3(log x)-2/3 for sufficiently large choices of the implied constants. Roth then gave an elementary proof that h = x 1/4+ ∈ is admissible, and by applying a result of van der Corput, he showed that one can take h≫ x 3/13 (log x)4/13Nair [6] later noted that the elementary proof could be modified to omit the ∈ in the exponent to get that h≫ x 1/4 is admissible, and more recently the first author [1] showed that one could obtain the result h≫x 3/13 by elementary means. Using further exponential sum techniques, Richert [8], Rankin [7], Schmidt [10], and Graham and Kolesnik [4] obtained the improvements h ≫ x 2/9log x, h = x θ +€ where θ = 0.221982…, θ = 109556/494419 = 0.221585…, and θ = 1057/4785 = 0.2208986…, respectively. The authors investigated the problem further.

The irreducibility of all but finitely many Bessel Polynomials

are all irreducible over the rationals and obtained several results concerning their irreducibility. The statement of this conjecture and his results are described in his book Bessel Polynomials [7]. The author in [4] established that almost all Bessel Polynomials are irreducible. More precisely, if k(t) denotes the number of n<.t for which y,~(x) is reducible, then k(t)=o(t). He later [5] observed that the argument could be strengthened to obtain k(t)<<t/logloglogt. More recently, it was shown by Sid Graham and the author [6] that a simplification of these methods with some additional elementary arguments lead to k(t)<<t 2/3. In this paper, we prove that yn(X) is irreducible for all but finitely many (possibly 0) positive integers n. Although the current methods lead to an effective bound on the number of reducible y,~(x), such a bound would be quite large and we do not concern ourselves with it. The coefficient of xJ in yn(x) is (2 +j) I-I~=1(2k-1) and, hence, integral. The constant term is 1. Thus, the irreducibility of yn(X) over the rationals is equivalent to the irreducibility of yn(x) over the integers. It is slightly more convenient to consider

On testing the divisibility of lacunary polynomials by cyclotomic polynomials

An algorithm is described that determines whether a given polynomial with integer coefficients has a cyclotomic factor. The algorithm is intended to be used for sparse polynomials given as a sequence of coefficient-exponent pairs. A running analysis shows that, for a fixed number of nonzero terms, the algorithm runs in polynomial time.

On the irreducibility of a certain class of Laguerre polynomials

R. Gow has investigated the problem of determining classical polynomials with Galois group Am, the alternating group on m letters, in the case that m is even (odd m being previously handled in work of I. Schur). He showed that the generalized Laguerre polynomial Lm(m)(x), defined below, has Galois group Am provided m>2 is even and Lm(m)(x) is irreducible (and obtained irreducibility in some cases). In this paper, we establish that Lm(m)(x) is irreducible for almost all m (and, hence, has Galois group Am for almost all even m).

On the factorization of consecutive integers

A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, or upon providing lower bounds for the number of prime divisors of such a product. As noted in [18], a basic technique in these arguments is to make a careful distinction between “small” and “large” primes, and then apply sophisticated results from multiplicative number theory. Along these lines, if we write ( n k ) = U · V, n ≥ 2k,

Short interval results for squarefree numbers

Abstract Elementary and exponential sum approaches are used to obtain results for gaps between squarefree numbers. An elementary proof is given that there exists a positive constant c such that if x is sufficiently large, then the interval (x, x + h], where h = cx 2 9 , contains a squarefree number. Using exponential sums, it is shown that for any e > 0 one can take h = x 47 217 + e .

On the distribution of gaps between squarefree numbers

Let s 1 , s 2 , … denote the squarefree numbers in ascending order. In [1], Erdős showed that, if 0 ≤ γ ≤ 2, then where B (γ) is a function only of γ. In 1973 Hooley [4] improved the range of validity of this result to 0 ≤ γ ≤ 3, and then later gained a further slight improvement by a method he outlined at the International Number Theory Symposium at Stillwater, Oklahoma in 1984. We have, however, independently obtained the better improvement that (1) holds for in contrast to the range derived by Hooley. The main purpose of this paper is to substantiate our new result. Professor Hooley has informed me that there are similarities between our methods as well as significant differences.

An elementary approach to short interval results for k-free numbers

Abstract An elementary proof is given that there exists a constant c2 = c2(k) such that for x sufficiently large (depending on k), the interval (x, x + c2x0], where θ = 5 (10k + 1) , contains a k-free number. This result improves on a previous result of Halberstam and Roth [J. London Math. Soc. (2) 26 (1951), 268–273] .

The irreducibility of almost all Bessel Polynomials

Abstract The irreducibility of the Bessel Polynomials yn(x) (described below) has been investigated by Emil Grosswald. He has obtained several interesting results on this subject; in particular, using his ideas, it is possible to prove that a positive proportion of the Bessel Polynomials are irreducible. This paper uses a different approach to deduce the stronger result that almost all Bessel Polynomials are irreducible.

The distance to an irreducible polynomial, II

P. Turan asked if there exists an absolute constant C such that for every polynomial f ∈ Z[x] there exists an irreducible polynomial g ∈ Z[x] with deg(g) ≤ deg(f) and L(f − g) ≤ C, where L(·) denotes the sum of the absolute values of the coefficients. We show that C = 5 suffices for all integer polynomials of degree at most 40 by investigating analogous questions in Fp[x] for small primes p. We also prove that a positive proportion of the polynomials in F2[x] have distance at least 4 to an arbitrary irreducible polynomial.