TASK #1 - Fortunate Numbers

Write a script to produce first 8 Fortunate Numbers (unique and sorted).

According to Wikipedia

A Fortunate number, named after Reo Fortune, is the smallest integer m >
1 such that, for a given positive integer n, pn# + m is a prime number,
where the primorial pn# is the product of the first n prime numbers.

Expected Output

3, 5, 7, 13, 17, 19, 23, 37

MY NOTES: ok.  Note that the Wikipedia article is not clear whether the
Nth Fortunate number is the smallest integer m>1 s.t. pN# + m is prime,
or whether that Nth value found is **A* Fortunate number, but the Fortunate
numbers themselves are sorted and have duplicates removed - and the Nth
Fortunate number is the Nth term of the sorted and deduped list.  I think
we want the latter, because the unsorted Fortunate numbers corresponding to
pn# for n=1..8 are 3, 5, 7, 13, 23, 17, 19, 23, i.e. they are not sorted and
contain a duplicate of 23.

Once you sort and dedup these, you get the desired sequence
3, 5, 7, 13, 17, 19, 23, 37

HOWEVER, it's not obvious how many unsorted Fortunate numbers we have
to calculate to know that we've found the first N sorted ones.  Let's keep
going until we have found N distinct Fortunate numbers and then sort them.
Is it possible for a later unsorted Fortunate number to be smaller
than one of the ones we've # found so far?  Yes, blast it:
The output of this program for N=1 is 3,5,7,13,17,19,23,37,61,67,71
and for N=12 is 3,5,7,13,17,19,23,37,47,61,67,71
(showing that the 12th distinct value found is 47, smaller than the 9th..11th
distinct value found)

Also, this program won't work for N>12 because the primorial values get so
enormous, like factorials.  I tried bigint but that slowed everything down
so much that I couldn't find the answers in the time I had available.


TASK #2 - Pisano Period

Write a script to find the period of the 3rd Pisano Period.

In number theory, the nth Pisano period, written as pi(n), is the period
with which the sequence of Fibonacci numbers taken modulo n repeats.

The Fibonacci numbers are the numbers in the integer sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, ...

For any integer n, the sequence of Fibonacci numbers F(i) taken modulo
n is periodic. The Pisano period, denoted pi(n), is the value of the
period of this sequence. For example, the sequence of Fibonacci numbers
modulo 3 begins:

0, 1, 1, 2, 0, 2, 2, 1,
0, 1, 1, 2, 0, 2, 2, 1,
0, 1, 1, 2, 0, 2, 2, 1, ...

This sequence has period 8, so pi(3) = 8.

MY NOTES: ok.  Once we've found a sequence of length L that immediately
repeats, i.e. that the first 2L entries comprise the same length L sequence
repeated twice, How do we know that this sequence repeats forever?