Problems & Puzzles: Puzzles

Puzzle 221. What are the next?

Regarding the following sequence of primes:

2, 7, 13, 43, 103, 1627, 25349, 315743, ...

1. Find the definition of the sequence

2. Find the next five terms

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 same basis:

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)




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