Shanta Laishram

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)

Stolarsky’s conjecture and the sum of digits of polynomial values

Let s q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 1978, Stolarsky showed that lim n→∞ inf s 2 (n 2 )/s 2 (n) = 0. He conjectured that, just as for n 2 , this limit infimum should be 0 for higher powers of n. We prove and generalize this conjecture showing that for any polynomial p(x) = a h x h + a h-1 x h-1 + ··· + a 0 ∈ ℤ[x] with h > 2 and a h > 0 and any base q, l im in f s q (p(n)) = 0. n→∞ s q (n) For any e > 0 we give a bound on the minimal n such that the ratio s q (p(n))/ s q (n) < e. Further, we give lower bounds for the number of n < N such that s q (p(n))/s q (n) < e.

On a conjecture on Ramanujan primes

For n ≥ 1, the nth Ramanujan prime is defined to be the smallest positive integer Rn with the property that if x ≥ Rn, then

Irreducibility of generalized Hermite-Laguerre Polynomials III

\pi(x)-\pi(\frac{x}{2})\ge n

GRIMM'S CONJECTURE ON CONSECUTIVE INTEGERS

where π(ν) is the number of primes not exceeding ν for any ν > 0 and ν ∈ ℝ. In this paper, we prove a conjecture of Sondow on upper bound for Ramanujan primes. An explicit bound of Ramanujan primes is also given. The proof uses explicit bounds of prime π and θ functions due to Dusart.

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

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.

THE SUM OF DIGITS OF n AND n 2

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.

Powers in products of terms of Pell's and Pell–Lucas Sequences

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

Some irreducibility results for truncated binomial expansions

Let

Extensions of Schur's irreducibility results

s_q(n)