T. N. Shorey

Pure powers in recurrence sequences and some related diophantine equations

Abstract We prove that there are only finitely many terms of a non-degenerate linear recurrence sequence which are q th powers of an integer subject to certain simple conditions on the roots of the associated characteristic polynomial of the recurrence sequence. Further we show by similar arguments that the Diophantine equation ax 2 t + bx t y + cy 2 + dx t + ey + f = 0 has only finitely many solutions in integers x , y , and t subject to the appropriate restrictions, and we also treat some related simultaneous Diophantine equations.

Transcendental infinite sums

Abstract We show that it follows from results on linear forms in logarithms of algebraic numbers such as where χ is any non-principal Dirichlet character and (Fn∞n=0 the Fibonacci sequence, are transcendental.

Almost Squares and Factorisations in Consecutive Integers

We show that there is no square other than 122 and 7202 such that it can be written as a product of k−1 integers out of k(≥3) consecutive positive integers. We give an extension of a theorem of Sylvester that a product of k consecutive integers each greater than k is divisible by a prime exceeding k.

Almost Squares in Arithmetic Progression

It is proved that a product of four or more terms of positive integers in arithmetic progression with common difference a prime power is never a square. More general results are given which completely solve (1.1) with gcd(n, d)=1, k≥3 and 1

Perfect powers in values of certain polynomials at integer points

1. For an integer v > 1, we define P(v) to be the greatest prime factor of v and we write P(1) = 1. Let m ≥ 0 and k ≥ 2 be integers. Let d 1 , ..., d t with t ≥ 2 be distinct integers in the interval [1, k]. For integers l ≥ 2, y > 0 and b > 0 with P(b) ≤ k, we consider the equation (m+d 1 )...(m+d t )=by 1 . (1) Put v1=1/2(1+1/l−2), l=4,5,... so that ½ 1 and kα i )≤k for 1 ≤ i ≤ t and hence t -1 k+π(k).

Number of prime divisors in a product of terms of an arithmetic progression

Let d ≥ 1, k ≥ 3, n ≥ 1 be integers with gcd(n, d) = 1. We denote ∆ = ∆(n, d, k) = n(n+ d) · · · (n+ (k − 1)d). For an integer ν > 1, we write ω(ν) and P (ν) for the number of distinct prime divisors of ν and the greatest prime factor of ν, respectively. Further we put ω(1) = 0 and P (1) = 1. For l coprime to d, we write π(ν, d, l) for the number of primes ≤ ν and congruent to l modulo d. Further, we denote by πd(ν) for the number of primes ≤ ν and coprime to d. The letter p always denote a prime number. Let W (∆) denote the number of terms in ∆ divisible by a prime > k. We observe that every prime exceeding k divides at most one term of ∆. Therefore we have W (∆) ≤ ω(∆)− πd(k). (1) If max(n, d) ≤ k, we see that n+(k−1)d ≤ k and therefore no term of ∆ is divisible by more than one prime exceeding k. Thus W (∆) = ω(∆)− πd(k) if max(n, d) ≤ k. (2)

Irreducibility of generalized Hermite-Laguerre Polynomials III

Abstract For a positive integer n and a real number α, the generalized Laguerre polynomials are defined by L n ( α ) ( x ) = ∑ j = 0 n ( n + α ) ( n − 1 + α ) ⋯ ( j + 1 + α ) ( − x ) j j ! ( n − j ) ! . These orthogonal polynomials are solutions to Laguerres Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties. He obtained irreducibility results of L n ( ± 1 2 ) ( x ) and L n ( ± 1 2 ) ( x 2 ) and derived that the Hermite polynomials H 2 n ( x ) and H 2 n + 1 ( x ) x are irreducible for each n. In this article, we extend Schurs result by showing that the family of Laguerre polynomials L n ( q ) ( x ) and L n ( q ) ( x d ) with q ∈ { ± 1 3 , ± 2 3 , ± 1 4 , ± 3 4 } , where d is the denominator of q, are irreducible for every n except when q = 1 4 , n = 2 where we give the complete factorization. In fact, we derive it from a more general result.

Contributions towards a conjecture of Erdos on perfect powers in arithmetic progressions

Let n, d, k ≥ 2, b, y and l ≥ 3 be positive integers with the greatest prime factor of b not exceeding k. It is proved that the equation n(n+d)... (n+d.(k−1)d) = by l has no solution if d exceeds d 1 , where d 1 equals 30 if l = 3; 950 if l = 4; 5 x 10 4 if l = 5 or 6; 10 8 if l = 7, 8, 9 or 10; 10 15 if l ≥11. This confirms a conjecture of Erdos on the above equation for a large number of values of d.

GRIMM'S CONJECTURE ON CONSECUTIVE INTEGERS

For positive integers n and k, it is possible to choose primes P1, P2,…, Pk such that Pi | (n + i) for 1 ≤ i ≤ k whenever n + 1, n + 2,…, n + k are all composites and n ≤ 1.9 × 1010. This provides a numerical verification of Grimms Conjecture.

The greatest prime divisor of a product of terms in an arithmetic progression

For an integer v > 1, we denote by co(v) and P(v) the number of distinct prime divisors of v and the greatest prime factor of v, respectively, and we put co(l) = O, P(1) = 1. Further we write zra(v) for the number of primes ~ k. Let d = 1. A well known theorem of Sylvester [15] states that