Anirban Mukhopadhyay


Research Summary:

For an integer $\nu>1$, we denote by $P(\nu)$ and $\omega(\nu)$ the greatest prime factor of $\nu$ and the number of distinct prime divisors of $\nu$, respectively. Further we put $P(1)=1$ and $\omega(1)=0$. Let $n,d,k,b,y$ be positive integers such that $b$ is square free, $d>1$, $k\ge 3$ and $P(b)\le k$. We consider the equation

\begin{displaymath} n(n+d)\cdots(n+(k-1)d)=by^2~~{\rm in}~~n,d,k,b,y~~{\rm with}~~P(b)\le k. \end{displaymath} (1)

Shorey and Tijdeman proved that with gcd$(n,d)=1$, (1) implies that $k$ is bounded by an effectively computable number depending only on $\omega(d)$. Further Shorey showed that the assumption gcd$(n,d)=1$ can be relaxed to $d\vert\!\!/ n$ in the preceding result. On the other hand, we observe that (1) may have infinitely many solutions in the case $d\vert n$. Next Saradha and Shorey showed that (1) with $b=1$ and $k\ge 4$ is not possible whenever $\omega(d)=1$. It has also been shown by Saradha and Shorey that (1) with $P(b)<k$, $d\vert\!\!/ n$, $\omega(d)=1$ and $k\ge 10$ does not hold. In collaboration with T. N. Shorey I studied (1) for $4\le k\le 9$, $P(b)<k$ and $7\le k\le 29$, $P(b)=k$ and obtained that only solution under these restrictions is $n=75,d=23,k=4,b=6,y=4620$.

Publications:

  1. (with Yong-Gao Chen) ; The view-obstruction problem for polygons;
    Publ. Math. Debrecen 60 (2002), no. 1-2, 101-105.
  2. (with S. D. Adhikari and G. Coppola); On the average of the
    sum-of-$p$-prime-divisors function; Acta Arith. 101 (2002), no. 4, 333-338.

Preprints:

  1. (with T. N. Shorey); Almost squares in arithmetic progression(II).
  2. (with T. N. Shorey) ; Square free part of products of consecutive integers.

Visits to other Institutes:

Visited TIFR in February and April, 2002.


Other Activities:

Two lectures in Basic Notion seminars on ``Congruence properties of Partition function''.

 



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