As far as I can say, there is no memory leak in your program. It is just using a lot of memory if $i and $j become larger than simply 4 or 5 because is is recursing very deeply.

The quick way which might fix this program would be to replace:

use strict;
print "what is the value of i:";
[download]

by:

use strict;
use warnings; # it would have warned you about deep recursion
use Memoize;
memoize ('ackerman');
print "what is the value of i:";
[download]

BTW, your function is not the standard Ackermann function as I know it. Using a recursive form of the standard Ackermann function on values 3 and 5 (result = 253), the function is called 42438 times; with memoize, it is called only 636 times. With 3 and 6, the memoized vesion is called 1277 times, the non-memoized version 172233. But even with memoize, don't call it with large numbers.