Problems & Puzzles:
Puzzle 221. What are the
Regarding the following sequence of
2, 7, 13, 43, 103,
1627, 25349, 315743, ...
1. Find the
definition of the sequence
2. Find the next
3. Can you argue if
this sequence is finite or infinite?
Johan Wiesenbauer and J. K.
Andersen got, independently, the next member of this sequence:
7338823. Both also argued that this sequence must be infinite, using the
The sequence is clearly infinite. As there are
arbitrary large prime gaps (remember that that for any positive integer n
the numbers n!+2,n!+3,...,n!+n are all composite) it suffices to say that
in order to continue the sequence p_1,..,p_n there is always a prime q
immediately before a sufficiently large prime gap such that all sums
mentioned above are composite. Among all those primes simply choose the
smallest one (Wiesenbauer)
The sequence is infinite. Proof: There are prime
gaps of arbitrary size, e.g. n!+2 to n!+n are all composites for n>1 since
k divides n!+k for k<=n. Let s(t) be the sum of the first t terms in the
sequence. Let p be a prime followed by a prime gap greater than s(t). Then
p satisfies the conditions of the sequence, possibly except minimality.
Either p or a smaller prime is term t+1 in the sequence and the
sequence cannot end (Andersen)