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-   -   Primes in n-fibonacci sequence and n-step fibonacci sequence (https://www.mersenneforum.org/showthread.php?t=21814)

sweety439 2016-12-07 17:43

Primes in n-fibonacci sequence and n-step fibonacci sequence
 
n-fibonacci sequence:

n OEIS sequence
1 [URL="http://oeis.org/A000045"]A000045[/URL]
2 [URL="http://oeis.org/A000129"]A000129[/URL]
3 [URL="http://oeis.org/A006190"]A006190[/URL]
4 [URL="http://oeis.org/A001076"]A001076[/URL]
5 [URL="http://oeis.org/A052918"]A052918[/URL]
6 [URL="http://oeis.org/A005668"]A005668[/URL]
7 [URL="http://oeis.org/A054413"]A054413[/URL]
8 [URL="http://oeis.org/A041025"]A041025[/URL]
9 [URL="http://oeis.org/A099371"]A099371[/URL]
10 [URL="http://oeis.org/A041041"]A041041[/URL]
11 [URL="http://oeis.org/A049666"]A049666[/URL]
12 [URL="http://oeis.org/A041061"]A041061[/URL]

n-step fibonacci sequence:

n OEIS sequence
1 [URL="http://oeis.org/A000012"]A000012[/URL]
2 [URL="http://oeis.org/A000045"]A000045[/URL]
3 [URL="http://oeis.org/A000213"]A000213[/URL]
4 [URL="http://oeis.org/A000288"]A000288[/URL]
5 [URL="http://oeis.org/A000322"]A000322[/URL]
6 [URL="http://oeis.org/A000383"]A000383[/URL]
7 [URL="http://oeis.org/A060455"]A060455[/URL]
8 [URL="http://oeis.org/A123526"]A123526[/URL]
9 [URL="http://oeis.org/A127193"]A127193[/URL]
10 [URL="http://oeis.org/A127194"]A127194[/URL]
11 [URL="http://oeis.org/A168083"]A168083[/URL]
12 [URL="http://oeis.org/A207539"]A207539[/URL]

Is there a project of searching primes in these sequences?

rogue 2016-12-07 18:03

I suggest that you take a look at MathWorld. If one exists, you would like find out about it there.

science_man_88 2016-12-07 18:47

[QUOTE=sweety439;448674]
Is there a project of searching primes in these sequences?[/QUOTE]

have you read up on recursive relations and parity arguments etc before posting these because with that and modular arithmetic on polynomials under the polynomial remainder theorem I bet you could do a quick scan of them first yourself.


for example we know things like:

any polynomial without a certain number of odd coefficients including the constant term have certain properties like always being even or switching back and forth etc. just based on parity arguments we can say things like:

any polynomial with an even number of odd coefficients will pair those up under half the integer x values. any with an odd number of odd coefficients including the constant term will be odd at least half the time.

we know by the pigeonhole principle that given modular remainders only can be 0 to n-1 ( n values) mod n that every n terms in a sequence has the same modular remainder mod n. etc.

for the relationship [TEX]a_n=a_{n-1}+a_{n-2}[/TEX] we have the obvious statements like unless the two values you sum are opposite parity then the nth value will be even. since the only even prime is 2 it makes it hard to be prime and have this occur.

sweety439 2017-02-01 16:02

Primes in Lucas sequences
 
Are there any research for primes in Lucas U(P, Q) and V(P, Q) sequences? i.e.

a(0)=0, a(1)=1, a(n+2)=P*a(n+1)-Q*a(n) for all n>=0

and

a(0)=2, a(1)=P, a(n+2)=P*a(n+1)-Q*a(n) for all n>=0

science_man_88 2017-02-01 16:48

[QUOTE=sweety439;451972]Are there any research for primes in Lucas U(P, Q) and V(P, Q) sequences? i.e.

a(0)=0, a(1)=1, a(n+2)=P*a(n+1)-Q*a(n) for all n>=0

and

a(0)=2, a(1)=P, a(n+2)=P*a(n+1)-Q*a(n) for all n>=0[/QUOTE]

see [url]http://mersenneforum.org/showpost.php?p=451977&postcount=13[/url] for an answer.

R. Gerbicz 2017-02-01 19:56

[QUOTE=sweety439;448674]
n-step fibonacci sequence:

n OEIS sequence
1 [URL="http://oeis.org/A000012"]A000012[/URL]
[...]
Is there a project of searching primes in these sequences?[/QUOTE]

Do not search primes in A000012.

LaurV 2017-02-02 13:35

[QUOTE=R. Gerbicz;451983]Do not search primes in A000012.[/QUOTE]
LOL Robert, good spot, haha.. .
:rofl: :w00t:

sweety439 2017-02-02 17:13

... of course... A000012 contains no primes since it only contains 1 ... XDDD

sweety439 2017-02-02 17:19

[QUOTE=sweety439;448674]n-fibonacci sequence:

n OEIS sequence
1 [URL="http://oeis.org/A000045"]A000045[/URL]
2 [URL="http://oeis.org/A000129"]A000129[/URL]
3 [URL="http://oeis.org/A006190"]A006190[/URL]
4 [URL="http://oeis.org/A001076"]A001076[/URL]
5 [URL="http://oeis.org/A052918"]A052918[/URL]
6 [URL="http://oeis.org/A005668"]A005668[/URL]
7 [URL="http://oeis.org/A054413"]A054413[/URL]
8 [URL="http://oeis.org/A041025"]A041025[/URL]
9 [URL="http://oeis.org/A099371"]A099371[/URL]
10 [URL="http://oeis.org/A041041"]A041041[/URL]
11 [URL="http://oeis.org/A049666"]A049666[/URL]
12 [URL="http://oeis.org/A041061"]A041061[/URL]

n-step fibonacci sequence:

n OEIS sequence
1 [URL="http://oeis.org/A000012"]A000012[/URL]
2 [URL="http://oeis.org/A000045"]A000045[/URL]
3 [URL="http://oeis.org/A000213"]A000213[/URL]
4 [URL="http://oeis.org/A000288"]A000288[/URL]
5 [URL="http://oeis.org/A000322"]A000322[/URL]
6 [URL="http://oeis.org/A000383"]A000383[/URL]
7 [URL="http://oeis.org/A060455"]A060455[/URL]
8 [URL="http://oeis.org/A123526"]A123526[/URL]
9 [URL="http://oeis.org/A127193"]A127193[/URL]
10 [URL="http://oeis.org/A127194"]A127194[/URL]
11 [URL="http://oeis.org/A168083"]A168083[/URL]
12 [URL="http://oeis.org/A207539"]A207539[/URL]

Is there a project of searching primes in these sequences?[/QUOTE]

The 11-step fibonacci sequence should be [URL="http://oeis.org/A127624"]A127624[/URL], not A168083.
[URL="http://oeis.org/A127624"] [/URL]

science_man_88 2017-02-02 17:29

[QUOTE=sweety439;452070]... of course... A000012 contains no primes since it only contains 1 ... XDDD[/QUOTE]

and if you do the math you'll see that primes could only occur in certain places in the others as well based on them being odd:

0,1,1,2,3,5,8,... notice a pattern
even+odd=odd
odd+odd=even so every third entry is even and can be eliminated from the search.( except 2)

1,1,1,3,5,9, they are all odd but technically could do other things to eliminate composites.
1,1,1,1,4,7,13,25,49,94,... every fifth number is eliminated because it's even.
1,1,1,1,1,.. all odd again.
1,1,1,1,1,1,6, every 7th number is eliminated because it's even
all odd again
every n+1th number is even and eliminated.

and that's just a start.

[url]https://en.wikipedia.org/wiki/Fibonacci_number#Primes_and_divisibility[/url] would help you with the fibonacci sequence.

sweety439 2017-02-03 06:18

[QUOTE=science_man_88;452075]and if you do the math you'll see that primes could only occur in certain places in the others as well based on them being odd:

0,1,1,2,3,5,8,... notice a pattern
even+odd=odd
odd+odd=even so every third entry is even and can be eliminated from the search.( except 2)

1,1,1,3,5,9, they are all odd but technically could do other things to eliminate composites.
1,1,1,1,4,7,13,25,49,94,... every fifth number is eliminated because it's even.
1,1,1,1,1,.. all odd again.
1,1,1,1,1,1,6, every 7th number is eliminated because it's even
all odd again
every n+1th number is even and eliminated.

and that's just a start.

[URL]https://en.wikipedia.org/wiki/Fibonacci_number#Primes_and_divisibility[/URL] would help you with the fibonacci sequence.[/QUOTE]

For n-fibonacci sequence, if the k-th term (F(n,k)) is prime (F(n,0)=0, F(n,1)=1, F(n,2)=n), then k must be prime, the only exception is F(1,4)=3.


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