Wikipedia:Reference desk/Archives/Mathematics/2023 September 19

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September 19

Primes "in the" primes, using triangular addition

The famous Cannonball problem in number theory, which has been solved using much more complicated knowledge from mathematical analysis, states that the sum of first N perfect squares are never again perfect squares except for only finite cases (namely, when N=1 and N=24). However, if the "perfect squares" are replaced by the "prime numbers", the "analogous" problem seems not to be true. I have (partly) constructed a sequence for the sum of first N prime numbers- the Nth terms in this sequence can be considered as a function of the index N, or g(N). One has g(N)>Nth prime for all N larger than 1 and g(N) is a monotonically increasing function. There are infinite number of even terms and also infinite number of odd terms in the sequence g(N) sandwiched between them (This can be proved easily by noting that 2 is the only even prime number). In the half-sequence of g(n) with the odd terms, one might looks for the prime rather than odd composite numbers. In fact, g(N) is a prime number for $N=1,2,4,12,14,60,64,\cdots$ corresponding to the value of $g(N)=2,5,17,197,281,7699,8893,\cdots$ (All prime-g(N) values after 2 result from the addition of all prime numbers from 2 to $3,7,37,43,281,311,\cdots$ respectively). The prime numbers in the sequence tends to be rarer as g(N) increases toward infinity - note the curious "large jump" from N=14 to N=60 (this distribution can be understood from the fact the larger the natural number, the lower the probability allowing it to be a prime). Claim: "The sequence g(N) contains infinite number of prime numbers, despite their almost-zero natural density among all the terms of g(N)". Is this a known result, or is it still an unproven hypothesis in the number theory, much like the strong Goldbach conjecture? The trivial but unusual coincidence-the number 281 and its value g(N)=7699 both occur in the sequence prime-g(N).2402:800:63BC:DB8D:D90A:C8AA:2A80:2CF7 (talk) 13:43, 19 September 2023 (UTC)

Based on what I've seen, questions involving the infiniteness of a set of primes fulfilling any criteria more than "they are prime" are, most of the time, not solved, unless there is some very obvious reason why such a set would be finite or empty. I would certainly bet that this is one of those unsolved questions. In any case, following the OEIS sequence for prime $g(N)$ reveals a 2018 paper by Romeo Meštrović on the subject which would indicate that the problem indeed does not have a resolution. GalacticShoe (talk) 14:28, 19 September 2023 (UTC)