Volume 7 - 1984


1. On the equation a(xm1)/(x1)=b(yn1)/(y1): II

Tarlok Nath Shorey.
Using the theory of linear forms in logarithms we generalize an earlier result with R. Balasubramanian on the equation of the title.

2. On the algebraic differential equations satisfied by some elliptic function I

P Chowla ; S Chowla.
When a is an odd positive integer it is implicit in the work of Jacobi that the functions Y=1σa(n)Xn where σa(n)=d/nda (the sum of the ath powers of the divisors of n) satisfy an algebraic differential equation; i.e., there is a polynomial T not identically 0, such that T(X,Y,Y1,,Ym)=0. When a=1 we give a new argument based on Ramanujan that we may take m=3 here.

3. On algebraic differential equations satisfied by some elliptic functions II

P Chowla ; S Chowla.
In (I) we obtained the ``implicit'' algebraic differential equation for the function defined by Y=1naxn1xn where a is an odd positive integer, and conjectured that there are no algebraic differential equations for the case when a is an even integer. In this note we obtain a simple proof that (this has been known for almost 200 years) Y=1xn2    (|x|<1)
satisfies an algebraic differential equation, and conjecture that Y=1xnk (where k is a positive bigger than 2) does not satisfy an algebraic differential equation.

4. A remark on goldbach's problem II

S Srinivasan.
In this paper, we give an alternative proof of a theorem of R. Balasubramanian and C. J. Mozzochi

5. On infinitude of primes

S Srinivasan.
Let K(>1) and k(>1) be given integers. In this paper we prove that eK(q)0modk[m] for infinitely many primes q, where m=ckloglogq for a certain ck>0 and eK(q) denotes the exponent of K modulo q. In particular, q1modk for infinitely many primes q.