Zeros of orthogonal polynomials

First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

On the order of magnitude of the difference between consecutive prime numbers

The objective of the present work was to study the blood pressure with respect to some anthropometric characteristics in adol The study comprised 1225 subjects (634 males and 591 females) aged 10-18 years. The range of average blood pressure (systolic/diastolic) was 116.33-134.52/ 78.23-92.62 mmHg in males and 117.71-128.75/ 77.97-88.85 mm Hg in females of 10 to 18 years of age respectively. It was observed that 11 and 12 years of age groups were more susceptible for pre-hypertension (4.49% and 4.24% respectively). Whereas, 10 years and 14 years age groups were more susceptible for hypertension (6.12% and 4.40% respectively). According to multivariate regression analysis, systolic and diastolic blood pressure was significantly associated with weight, BMI and waist circumference. The results of the present analysis support the hypothesis that BMI is a significant predictor of blood pressure in adolescent age groups.

List of Publications of

We prove that for any positive integers x,d and k with gcd (x,d)=1 and 3 11, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

/pdf/perfect-powers-from-products-of-consecutive-terms-in-5bwlgo712g.pdf

Perfect powers from products of consecutive terms in arithmetic progression

https://www.isid.ac.in/~shanta/PAPERS/IndagNoAP.pdf

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

It is proved that equation (1) with 4 ≤ k ≤ 109 does not hold. The paper contains analogous result for k ≤ 100 for more general equation (2) under certain restrictions.

http://www.math.tifr.res.in/~shorey/Eulerextn.pdf

An extension of a theorem of Euler

The conjecture of Masser-Oesterl\'e, popularly known as $abc$-conjecture have many consequences. We use an explicit version due to Baker to solve a number of conjectures.

Baker's explicit abc-conjecture and applications

In this paper, we consider the usual Pell and Pell–Lucas sequences. The Pell sequence is given by the recurrence un = 2un-1 + un-2 with initial condition u0 = 0, u1 = 1 and its associated Pell–Lucas sequence is given by the recurrence vn = 2vn-1 + vn-2 with initial condition v0 = 2, v1 = 2. Let n, d, k, y, m be positive integers with m ≥ 2, y ≥ 2 and gcd(n, d) = 1. We prove that the only solutions of the Diophantine equation unun+d⋯un+(k-1)d = ym are given by u7 = 132 and u1u7 = 132 and the equation vnvn+d⋯vn+(k-1)d = ym has no solution. In fact, we prove a more general result.

/pdf/powers-in-products-of-terms-of-pell-s-and-pell-lucas-1f0aivu2fo.pdf

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

For an integer ν > 1, we denote by ω(ν) and P (ν) the number of distinct prime divisors of ν and the greatest prime factor of ν, respectively, and let ω(1) = 0, P (1) = 1. Also we denote by W (∆′) the number of terms in ∆′ divisible by a prime greater than k. Let pi denote the ith prime number. Thus p1 = 2, p2 = 3, . . . . We shall always write p for a prime number. Since ∆ is divisible by k!, we observe that

/pdf/number-of-prime-divisors-in-a-product-of-consecutive-13dzr8lyod.pdf

Shanta Laishram

Papers

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

An extension of a theorem of Euler

Baker's explicit abc-conjecture and applications

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

Number of prime divisors in a product of consecutive integers