Well, this works:

    my $n = ...;
    my $b = ...;

    my @possibles;

    my $p = '1';
    while ($p <= $n) {
        push @possibles, ('1' x $p);
        $p *= $b;
    }

    my ($pat) =
        map qr/$_/,
        join '|',
        @possibles;

    print( "$n: ", ('1' x $n) =~ /^$pat\z/ ?1:0, "\n" );
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You can do almost all of the above inside the pattern, except for the $n <= $number check:

    my $n    = ...;
    my $b    = ...;
    my $b_m1 = $b-1;

    my $pat = qr/(1|((?-2))\g{-1}{$b_m1})/;

    print( "$n: ", ('1' x $n) =~ /^$pat\z/ ?1:0, "\n" );
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Without that check, the regex engine preemptively (and sometimes accurately) throws "Infinite recursion in regex".

Back to the drawing board.

Update: As best as I can tell, you'd need to be able to change the topic as follows:

    local our $pat;
    $pat = qr/
        1
    |
        (1+)
        \g{-1}{$b_m1}
        (?(?{ $^N =~ $pat })(?=)|(?!))
    /x;
[download]

But that deosn't even work *with* (?{})!