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#!/usr/bin/env perl -l

...
my $next_file = 'pm_1069813_results' . ++$highest . '.txt';

print "Next file: $next_file";

Next file: pm_1069813_results4.txt

$ ls -1 pm_1069813_results*.txt
pm_1069813_results1.txt
pm_1069813_results2.txt
pm_1069813_results3.txt