Titlebook: Number Theory and Discrete Mathematics; A. K. Agarwal,Bruce C. Berndt,Michel Waldschmidt Conference proceedings 2002 Springer Basel AG 200

extemporaneous

Irksome

brother

On a Conjecture of Andrews-II, The case . of the 1974 conjecture of Andrews on two partition functions ... and ... was proved by the first author and T.G. Sudha [On a conjecture of Andrews, Internat.J. Math. and Math. Sci. Vol. 16, No. 4 (1993), 763–774].In this paper we prove the two cases of . and . of the same conjecture.

Hirsutism

encyclopedia

A Report on Additive Complements of the Squares,This article provides an account of some recent investigations into the behaviour of additive complements of the sequence of squares. We begin by defining this notion.

GILD

Transcendental Infinite Sums and Some Related Questions,Erd?s and Chowla put forward some questions regarding non-vanishing of certain infinite sums. In this article, we present an expository account of results obtained in that direction. These include some interesting results of Baker, Birch and Wirsing and some recent work of the present author jointly with Saradha, Shorey and Tijdeman.

Mammal

,The Lehmer Problem on the Euler Totient: A Pendora’s Box of Unsolvable Problems,The celebrated seventy year old, innocent looking problem of D.H. Lehmer [.] asking for composite numbers, if any, satisfying the relation ?(.)| (. — 1), where ?(.) is the Euler totient, is still unsolved. This is easily seen to be equivalent to asking the . Given ., . odd and ?(.)| (. — 1), is . necessarily a prime?

爵士樂

The Problems Solved by Ramanujan in the Journal of the Indian Mathematical Society,Between 1912 and 1914, eight solutions by Ramanujan to questions posed in the . were published. Since these solutions have not heretofore appeared elsewhere, and since some of these problems evidently motivated certain entries in his notebooks [.], in this paper, we present all eight problems and solutions and provide some commentary on them.

conquer

A (Conjectural) 1/3-phenomenon for the Number of Rhombus Tilings of a Hexagon which Contain a Fixedith side lengths 2. +.,2. + ., .,. a, 2n + ., . contains the (horizontal) rhombus with coordinates (2n + x, 2n + y) is equal to .,where ..(n) . rational function in n. Several specific instances of this “1/3-phenomenon” are made explicit.

神化怪物

,Observations on Some Algebraic Equations Associated with Ramanujan’s Work,,which he solved by radicals,and the Diophantine equation x. + ..... 1, which appears in ., along with an astonishing solution. It is shown that, in general, the equation with 5 iterated square roots cannot be solved by radicals and that the Diophantine equation has solutions not previously quoted.