## Proof of less-fortunate numbers

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• Proof of less-fortunate numbers (as seen on http://primes.utm.edu/glossary/page.php?sort=FortunateNumber ): By observation: The largest prime of the solution
Message 1 of 1 , Feb 20, 2005
Proof of less-fortunate numbers (as seen on
http://primes.utm.edu/glossary/page.php?sort=FortunateNumber ):

By observation: The largest prime of the solution for each
primordial to "Carl Pomerance
Observation" by Said El Aidi on
http://www.primepuzzles.net/problems/prob_002.htm are less-fortunate
numbers.

Knowing the above fact, how big does the size of the gap to the next
prime, assuming with and without that p_n# (+/-) 1 is prime, have to
be in order to fail to be a Fortunate Number? Chris stated on the
page (I changed the notation) "So we are looking for a prime gap
near p_n# of about (log p_n# log log p_n#)^2." But two things are
for sure, there is a gap with a size >= p_n after p_n# and . (For
the same reason that there is a gap after p_n! >= p_n but with a
common multiple added to p_n#.) Is this the only thing that can be

Can a modification of Aidi's work be stated to prove a Fortunate
number, q, exist between p_n# + p_n < q < p_n# + (p_(n+1))^2, so
that r = q - p_n# with r being prime? Does knowing the status of
less-fortunate numbers satisfy a necessary or sufficient condition
for Fortunate number q? I am thinking that the number of primes in
the Aidi range is greater than this range but there is always one
prime because of the required solutions to the congruence of numbers
between p_n# and p_n! in the range p_n# + p_n < q < p_n# + (p_(n+1))
^2. Another words the solutions must be spread out, this small group
can not be the last grouping.

On a different note, I am rewriting Aidi's statements, I am hoping