Volume 7 - 1984

1. On the equation $a(x^m-1)/(x-1)=b(y^n-1)/(y-1)$: II

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

When $a$ is an odd positive integer it is implicit in the work of Jacobi that the functions $Y=\sum_1^{\infty} \sigma_a(n)X^n$ where $\sigma_a (n) = \sum_{d/n} d^a$ (the sum of the $a$th powers of the divisors of $n$) satisfy an algebraic differential equation; i.e., there is a polynomial $T$ not identically $0$, such that $T(X, Y, Y_1, \ldots, Y_m)=0$. 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

In (I) we obtained the implicit'' algebraic differential equation for the function defined by $Y=\sum_1^{\infty}\frac{n^a x^n}{1-x^n}$ 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) $Y=\sum_1^{\infty}x^{n^2}~~~~(\vert x\vert<1)$ satisfies an algebraic differential equation, and conjecture that $Y=\sum_1^{\infty} x^{n^k}$ (where $k$ is a positive bigger than $2$) does not satisfy an algebraic differential equation.

4. A remark on goldbach's problem II

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

5. On infinitude of primes

Let $K (>1)$ and $k (>1)$ be given integers. In this paper we prove that $e_K(q)\equiv0 \mod k^{[m]}$ for infinitely many primes $q$, where $m=c_k\log\log q$ for a certain $c_k>0$ and $e_K(q)$ denotes the exponent of $K$ modulo $q$. In particular, $q\equiv1 \mod k$ for infinitely many primes $q$.