Problems & Puzzles: Puzzles

Puzzle 923. 2*P+Q is prime Let's study here the sequence of primes produced by a recursive application of: 2*P+Q or 2*P-Q where Q is nextprime(P) or previousprime(P)   This mean that we will study separately the following four sequences produced by recursion: a) 2*P+nextprime(P) b) 2*P-nextprime(P) c) 2*P+previousprime(P) d) 2*P-previousprime(P) In each case we will ask for the largest recursive sequence that ends whenever the next term is composite.   Examples, for a) 2*P+nextprime(P):   K=2: 2, 7 K=3: 5, 17, 53 K=4: 739, 2221, 6679, 20047 K=5: 61871, 185621, 556883, 1670657, 5011973 K=6: 1538419, ... , 373837459   Examples, for b) 2*P-nextprime(P):   K=1: 2 K=2: 5, 3 K=3: 53, 47, 41 ... K=6: 24930001, ... 24929911   Q. Can you send your largest sequence for each of these four defined above?

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Contributions came from Jan van Delden

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Jan wrote on May 1, 2018:

2P+nextprime(P), k=8

27263864713, 81791594149, 245374782451, 736124347369,

2208373042111, 6625119126463, 19875357379417, 59626072138267

 

2P-nextprime(P), k=8
2575726851211, 2575726851169, 2575726851127, 2575726851097,

2575726851067, 2575726851037, 2575726851007, 2575726851001

 

2P+previousprime(P), k=9

1904076904427, 5712230713271, 17136692139779, 51410076419267, 154230229257773,

462690687773297, 1388072063319833, 4164216189959453, 12492648569878283

 

2P-previousprime(P), k=8
1500308876993, 1500308877047, 1500308877077, 1500308877107,

1500308877131, 1500308877149, 1500308877167, 1500308877173

 

With the -sign I searched until 1000*2^32. With the +sign I stopped the routine earlier.

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