I haven't understood your problem, but there's a lot of things about your style that are better done different, and some that outright need fixing. So I'll just take notes along the way as I'm cleaning it up.

First thing I notice:

@a=@c=@t=@g=@n=@a2=@c2=@t2=@g2=@n2=();
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(@a, @c, @t, @g, @n, @a2, @c2, @t2, @g2, @n2) = ();
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for (my $k=0;$k<$number_positions; $k++) {
    my (@a,@c,@t,@g,@n,@a2,@c2,@t2,@g2,@n2);
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if ($letter1=~ /[aA]/) { ... }
...
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if (uc $letter1 eq 'A') { ... }
if (uc $letter1 eq 'C') { ... }
if (uc $letter1 eq 'T') { ... }
if (uc $letter1 eq 'G') { ... }
# ...
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$letter1 = uc $letter1;

if ($letter1 eq 'A') { ... }
if ($letter1 eq 'C') { ... }
if ($letter1 eq 'T') { ... }
# ...
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for (my $k=0;$k<$number_positions; $k++) {
    my (%base1, %base2);
    # ...
        $letter1 = uc $letter1;
        $letter2 = uc $letter2;

        push @${$base1{$letter1}}, $letter1;
        push @${$base2{$letter2}}, $letter2;
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for (my $k=0;$k<$number_positions; $k++) {
    my (%base1, %base2);
    # ...
        $base1{uc $letter1}++;
        $base2{uc $letter2}++;
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Now, the $number_* variables were already crying out to become elements of a %number hash. But we've already taken care of that with the %base counter hashes.

Next we have stuff like

"$number_a" + "$number_c" + "$number_t" + "$number_g" + "$number_n"
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$number_a + $number_c + $number_t + $number_g + $number_n
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$base1{A} + $base1{C} + $base1{T} + $base1{G} + $base1{N}
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Also note how you're doing this calculation over and over and over. Just do it once!

my $total1 = $base1{A} + $base1{C} + $base1{T} + $base1{G} + $base1{N}
+;
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my %percent1 = (
    A => ($base{A} * 100) / $total1,
    C => ($base{C} * 100) / $total1,
    T => ($base{T} * 100) / $total1,
    G => ($base{G} * 100) / $total1,
    N => ($base{N} * 100) / $total1,
);
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my %percent1 = map { $_ => ( $base{$_} * 100) / $total1 } 'A', 'C', 'T
+', 'G', 'N';
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my %percent1 = map { $_ => ( $base1{$_} * 100) / $total1 } qw(A C T G 
+N);
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#!/usr/bin/perl

use warnings;
use strict;

chomp(my $input = <STDIN>);
open(INPUT, "$input") or die "Cannot open file: $!";

chomp(my $number_positions = <STDIN>);

chomp(my $patient_number = <STDIN>);

chomp(my $output = <STDIN>);
open(OUTPUT, ">$output.txt") or die "Cannot open file: $!";

my @prettybase = <INPUT>;
close(INPUT);

for my $k (0 .. $number_positions - 1) {
    my (%base1, %base2);

    for my $i (0 .. $patient_number) {
        my ($position, $patient, $letter1, $letter2) =
            split (/\t/, $prettybase[$i]);

        $base1{uc $letter1}++;
        $base2{uc $letter2}++;
    }

    my $total1 = $base1{A} + $base1{C} + $base1{T} + $base1{G} + $base
+1{N};
    my %percent1 = map { $_ => ( $base1{$_} * 100) / $total1 } qw(A C 
+T G N);

    my $total2 = $base2{A} + $base2{C} + $base2{T} + $base2{G} + $base
+2{N};
    my %percent2 = map { $_ => ( $base2{$_} * 100) / $total2 } qw(A C 
+T G N);

    print OUTPUT "Position\#\: $patient\n";
    print OUTPUT "\tAllele 1  Allele 1 percentage\n";
    for(qw(A C T G N)) {
        print OUTPUT "$_:\t$base1{$_}\t$percent1{$_}\%\n\n";
    }
    print OUTPUT "\tAllele 2  Allele 2 percentage\n";
    for(qw(A C T G N)) {
        print OUTPUT "$_:\t$base2{$_}\t$percent2{$_}\%\n\n";
    }
}
close(OUTPUT);
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Aristotle, all I can say is wow. Thanks so much for that...I just learned a tremendous amount, and I realize that I need to learn to pare down all my variables to make things cleaner.
Bioinformatics

##
## Will process the records between $start and $end
## and count the occurances of the third
## field, returning the total count and a 
## has containing the individual counts keyed
## by the letter.  
##
sub processRec {
  my($arrayRef, $start, $end) = @_ ;
  my($i, $total, $letter_counts) ;
  my($position, $patient, $letter1, $letter2) ;
  
  for( $i = $start ; $i <= $end ; $i++ ) {
    ($position, $patient, $letter1, $letter2)=split(/\t/, $arrayRef->[
+$i]);

    $letter_counts->{uc $letter1} += 1 ;
  }
  $total = 0 ;
  for( values %$letter_counts ) {
    $total += $_ ;
  } # for

  return($total, $letter_counts) ;
}

@prettybase = map chomp, <INPUT> ;

#
# do something to determine start and stop points
#
($total, $counts) = processRec(\@prettybase, start, end) ;

while( ($letter, $count) = each %$counts ) {
  print "$letter => ", ($count*100)/$total, "\n" ;
}
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Yes, that is correct. Technically, I could have used pattern matching rather than a counter to go through @prettybase, but normally a counter works well for these type of programs. Unfortunately, not in this case.
Bioinformatics

my @prettybase;

my $seen = 0;
while (<PRETTYBASE>)
{
    chomp;
    my @info = split /\t/;

    unless ($info[1] == $patient_number)
    {
        # Use this to short-cut out of the file once we've found our p
+atient info.
        last if $seen;
        next;
    }

    $seen = 1;

    push @prettybase, $_;
}

# At this point, everything in @prettybase corresponds to the patient 
+specified.
# Modify your algorithms accordingly.
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------
We are the carpenters and bricklayers of the Information Age.

The idea is a little like C++ templates, except not quite so brain-meltingly complicated. -- TheDamian, Exegesis 6

... strings and arrays will suffice. As they are easily available as native data types in any sane language, ... - blokhead, speaking on evolutionary algorithms

Please remember that I'm crufty and crochety. All opinions are purely mine and all code is untested, unless otherwise specified.