You would put one of the constructs suggested by haukex or LanX inside the while from your original code, around your if/else block.

(Also, rather than having that if/else block, and repeating two almost-identical pieces of code: after generating the two random numbers, I would suggest using this to swap them, so the first is always bigger than the second:

if($random_integer1 < $random_integer2) 
{ 
    my $tmp = $random_integer2; 
    $random_integer2 = $random_integer1; 
    $random_integer1 = $tmp;
}
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This is perfectly OK, but the temporary variable is not really needed and this can be simplified as follows:

($random_integer1, $random_integer2) = ($random_integer2, $random_inte
+ger1) if $random_integer1 < $random_integer2;
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that was exactly one the "more perlish ways" I was referencing. I would have done it that way for myself, but I thought the explicit swap was more concrete for the OP to understand what was going on.

Do I need to store the random integers in variables or something.?

Yes, but you are already doing that. Here's one approach with a subroutine and using an array to hold the operands.

#!/usr/bin/env perl 

use strict;
use warnings;


while (1) {
    my @operands = (int(rand(100)), int(rand(100)));
    ask ($operands[0] > $operands[1], @operands);
}

sub ask {
    my ($subtract, @args) = @_;
    my $input;

    my $answer = $subtract ? $args[0] - $args[1] : $args[0] + $args[1]
+;
    my $oper   = $subtract ? '-' : '+';

    until (defined $input && $input == $answer) {
        print "Nope, try again\n" if defined $input;
        print "What is $args[0] $oper $args[1] ? ";
        chomp( $input = <STDIN> );
    };
    print "Success!\n";
}
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Edit: sigil fix on the ask() call.

wow.! this works great. Thank you. However, it appears rather complicated for me to understand and therefore learn from. Can I be cheeky and ask what the code is doing line per line.? No worries if you can't.

Well, I don't want to over-explain since the best way to learn is often to figure things out for yourself. However, let's examine what I suppose could be the 2 most confusing parts.

ask ($operands[0] > $operands[1], @operands);
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This is a subroutine call. You'll see we've defined the subroutine ask in the script? This is how it is called. There are either 2 or 3 arguments inside the brackets depending on how you see things. The first one is $operands[0] > $operands[1] which is a logical expression and therefore either true or false and the second one is @operands which is an array with the 2 random values previously assigned. Note that originally this line had an @ symbol at the start of $operands[0] which although OK was misleading (and a typo) so I've corrected it now.

my $answer = $subtract ? $args[0] - $args[1] : $args[0] + $args[1];
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This is a variable assignment which computes the answer in one way or another depending on the logical value in $subtract which we passed into the sub as the first argument. It uses the ternary operator which is of the form condition ? eval-if-true : eval-if-false. Think of it as a short-hand way of writing an if-else block.

The rest of the code is just loops and prints, really. See if you can trace through the logic now.