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?

Solution:

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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