Problems & Puzzles: Puzzles

 Puzzle 756. Primonacci Dimitry Kamenetsky sent the following nice puzzle. Let S be a sequence of prime numbers such that S(1)=a, S(2)=b, S(n+1)=S(n)+S(n-1)+c, where a0, for 1<=n<=k  Here is my best solution for this puzzle. It is a sequence with length 19: 10331, 67073, 109619, 208907, 350741, 591863, 974819, 1598897, 2605931, 4237043, 6875189, 11144447, 18051851, 29228513, 47312579, 76573307, 123918101, 200523623, 324473939 Q. Find a larger sequence like this.

Contribution came from Emmanuel Vantieghem

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Emmanuel wrote:

It was not possible for me to obtain a significant contribution to puzzle 756.
But, a very intersting related  problem might be the following :

Find odd nubers  a, b, c  such that the sequence  ( F(n) )  with
F(1) = a, F(2) = b, F(n) = F(n-1) + F(n-2) + c, contains no prime at all.
(I ommitted the condition  a < b  because it is completely unnecessary:
if you find a solution   F(n)  with  a > b, you can take the sequence  F*(n) defined by  F*(n) = F(n+1) )

That problem is somewhat analogue with  problem 31.  Finding a decent covering could solve it, but the fact that all numbers in the sequence must be odd makes it very very difficult.
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Jarek Biszczuk wrote on Set 24, 2014:

Problem is the same:
Find maximum k where sequence F(n+3)=2*F(n+2)-F(n),
F(1)<F(2)<F(1)+F(2)<F(3)
1<=n<=k, F(n) is prime.

Original problem:
c=F(3)-F(1)-F(2)
s(1)=F(1); s(2)=F(2), s(n)=s(n-1)+s(n-2)+c

My observation
(0 != F(1)=F(2)=F(3) mod 3)
and[ (0 != F(1)=F(2)=F(3) mod 5) or (F(1)=x, F(2)=y, F(3)=z where x,y,z   cyclic values from (1,3,4,2) e.g.  x=4,y=2,z=1)]
and (0 != F(1)=F(2)=F(3) mod 7).

I found sequences:
11<=F(1)<F(2)<F(3)< 570000
the condition F(1)+F(2) < F(3) is omitted:

k, F(1), F(2), F(3)
21, 145721, 190283, 195869
20,  74363, 170669, 202127
19, 108089, 190829, 495749 **
20, 127103, 129959, 196277
19, 229423, 244669, 258067
20*,309391, 529933, 539509
20, 195527, 307121, 318503
19, 188681, 208253, 409349 **
20, 127103, 129959, 196277
19,  72503, 217823, 358313 **
19, 354323, 473519, 580787
19, 211543, 354469, 389287
19, 173617, 487651, 511633
20, 129491, 136883, 235289
19*, 10331,   67073, 109619 **
19*,175811, 401813, 439949
19,   55313, 311303, 428693 **
20*,217333, 449929, 485167
19, 425149, 505537, 559591
19,   58393,  62299, 445507 **
19,  322237, 340381, 522073
19,  81929,  142619, 395459 **

* F(0)=2*F(2)-F(3) is prime, and F(0)>F(1)
** F(1)+F(2)<F(3)
A few days later he added:

I found another numerical sequences, one of them has k = 20 and F (1) + F (2) <F (3) as requested by puzzles.

11<=F(1)<F(2)<F(3)<700 000
k,F(1), F(2), F(3)
20, 45613, 488419, 664777 **

19,188827,468841,633943
21,474847,534571,691693
19*,457621, 617053, 674929
19, 339091, 505201, 712051
19, 351731, 554843, 612719
19, 136277,149297, 599537 **
19***, 464561, 632393, 684599
19*, 39323, 608633, 696833 **
20, 54269, 592919, 644159
19, 571037,615767,61744

* F(0)=2*F(2)-F(3) is prime, and F(0)>F(1)
** F(1)+F(2)<F(3)
*** F(0)=580187, F(-1)=296729 are prime

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