Problems & Puzzles: Puzzles

 Puzzle 464. p+4.x2 Farideh sent the following nice puzzle: The following sequence (A145741) of primes gives numbers p such that p+4*m^2 for m=0,1,2,...,10 are prime. 163, 409333, 1867783, 828497443, 2059599067, ? Q1. What is the next term? A very strange thing related to the first term that we found before (see the answer to puzzle 461) is : 163+4*m^2 for m=0,1,2,...,19 are prime (no other prime like 163 up to 3.10^10) Q2. Is there another number like this one?

Andersen wrote:

Q1. There are 3136 p values below 10^15. The next 10 are: 5767711867,
8965599883, 9655686727, 9671300983, 12627473917, 19635778453,
31428255373, 76292280733, 140192390683, 141077258953.

Here is a 50-digit value: p = 7105935*107#+178661021053 =
18236501877182422164330080901948552029012619200503.

Q2. p=163 is the only prime below 10^17 such that p+4*m^2 is prime for
m = 0 to 16. The next p which goes all the way to m=19 is probably much
larger, but there should be infinitely many based on common conjectures.

A092120(n) is the smallest p such that p+4*m^2 is prime for m = 0 to n,
but composite for m = n+1. The 10 currently listed terms are:
19, 3, 227, 43, 53593, 7, 67, 37, 1483087, 1867783.
(227 should have been 277)
I will submit term 11 to 15: 9671300983, 376040154163, 13491637509487,
604490757900187, 409333.
All later terms are above 10^17 (assuming they exist), except the 19th term
which is 163.

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