Problems & Puzzles: Puzzles

 Puzzle 1012. Primes on the skin of Fibonacci's Here we will recall your attention to these primes that are as close at they can be to Fibonacci integers. These come in three forms:   1) If F(i)=even, then Primes can be F(I)+/-1 2) If F(i)=odd, then Primes can be F(I)+/-2 3) Special cases: If F(I)=odd, then Primes can be F(I)+/-1: F(4)-1=3-1=2 and (F(1) & F(2))+1=1+1=2   Here are some primes of the first two classes that I have computed until the capacity of my old Ubasic.   1) For F(i)=even, F(I)+/-1 F( 3 )+1=2+1=3 Prime F( 6 )-1=8-1=7 Prime As a matter of fact Emmanuel Vantieghem has stated that these are the only possible, in a comment to A000045.   Q1. Can you get some more, or prove that there are no more than these two?     2.a) For F(i)=odd, F(I)-2= Prime F( 5 )-2=Prime (Cousin) F( 7 )-2=Prime F( 8 )-2=Prime (Cousin) F( 10 )-2=Prime F( 20 )-2=Prime F( 100 )-2=Prime F( 106 )-2=Prime F( 508 )-2=Prime F( 586 )-2=Prime F( 938 )-2=Prime F( 1162 )-2=Prime F( 1244 )-2=Prime F( 1444 )-2=Prime F( 1492 )-2=Prime F( 2884 )-2=Prime F( 3124 )-2=Prime F( 3170 )-2=Prime F( 4564 )-2=Prime   2.b) For F(i)=odd, F(I)+2= Prime F( 4 )+2=prime F( 5 )+2=Prime (Cousin) F( 8 )+2=Prime (Cousin) F( 14 )+2=Prime F( 22 )+2=Prime F( 40 )+2=Prime F( 76 )+2=Prime F( 88 )+2=Prime F( 158 )+2=Prime F( 196 )+2=Prime F( 230 )+2=Prime F( 236 )+2=Prime F( 260 )+2=Prime F( 788 )+2=Prime F( 800 )+2=Prime F( 1280 )+2=Prime F( 2374 )+2=Prime F( 2774 )+2=Prime Q2. Can you get other Fibonacci integers as F(5) & F(8), sandwiched in the middle of two cousin primes, or prove that there are no more?   Q3. Can you extend these two list 2.a) and 2.b)?

During the week 8-15 August, 2020, contributions came from Oscar Volpatti, Adam Stinchcombe, Simon Cavegn, Emmanuel Vantieghem, Paul Cleary.

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Oscar wrote:

Q1
Emmanuel Vantieghem's claim is correct.

A Fibonacci number F(i) is even iff i = 3*j.
All numbers F(3*j)-1 and F(3*j)+1 can be factored, as follows:
1) F(12*q+3  ) - 1 = F(6*q+2) * (F(6*q+2) + F(6*q))
2) F(12*q+6  ) - 1 = F(6*q+2) * (F(6*q+5) + F(6*q+3))
3) F(12*q+9  ) - 1 = F(6*q+4) * (F(6*q+6) + F(6*q+4))
4) F(12*q+12) - 1 = F(6*q+7) * (F(6*q+6) + F(6*q+4))
5) F(12*q+3  ) + 1 = F(6*q+1) * (F(6*q+3) + F(6*q+1))
6) F(12*q+6  ) + 1 = F(6*q+4) * (F(6*q+3) + F(6*q+1))
7) F(12*q+9  ) + 1 = F(6*q+5) * (F(6*q+5) + F(6*q+3))
8) F(12*q+12) + 1 = F(6*q+5) * (F(6*q+8) + F(6*q+6))

These relations can be proven by adding the base identities:
F(m)*F(n) + F(m+1)*F(n+1) = F(m+n+1)
F(k+1)*F(k-1) - F(k)^2 = (-1)^k
for a suitable choice of values m, n, k:
1) m = n = k = 6*q+1;
2) m = 6*q+2, n = k = 6*q+3;
3) m = n = 6*q+4, k = 6*q+5;
4) m = 6*q+4, n = k = 6*q+7;
5) m = n = 6*q+1, k = 6*q+2;
6) m = 6*q+1, n = k = 6*q+4;
7) m = n = k = 6*q+4;
8) m = 6*q+5, n = k = 6*q+6;

As an example:
F(12*q+12) + 1 =
(F(6*q+5)*F(6*q+6) + F(6*q+6)*F(6*q+7))  +  (F(6*q+5)*F(6*q+7) - F(6*q+6)^2) =
F(6*q+5)*(F(6*q+6) + F(6*q+7))  +  F(6*q+6)*(F(6*q+7) - F(6*q+6)) =

F(6*q+5)*F(6*q+8) + F(6*q+6)*F(6*q+5) =
F(6*q+5)*(F(6*q+8) + F(6*q+6)), qed.

List 1.a
F(3)-1 = 1*(1+0) = 1, not prime;
F(6)-1 = 1*(5+2) = 7, prime;

for j>2, both factors become greater than 1, so that F(3*j)-1 is composite:
F(9)-1 = 3*(8+3);
F(12)-1 = 13*(8+3);
and so on.

List 1.b
F(3)+1 = 1*(2+1) = 3, prime;
for j>1, both factors become greater than 1, so that F(3*j)+1 is composite:
F(6)+1 = 3*(2+1);
F(9)+1 = 5*(5+2);
F(12)+1 = 5*(21+8);
and so on.

Q2
I found no more pairs of cousin primes F(i)-2, F(i)+2 for 8 < i < 10^5.
Larger pairs are not impossible, but they are unlikely.

Q3
List 2.a, entries with 4564 < i < 10^5
...
F(6968)-2 is prime (1456 digits)

F(10042)-2 is prime (2099 digits)
F(11720)-2 is prime (2449 digits)
F(12650)-2 is prime (2644 digits)
F(15604)-2 is PRP (3261 digits)
F(15820)-2 is PRP (3306 digits)
F(27290)-2 is PRP (5703 digits)
F(28960)-2 is PRP (6052 digits)
F(32090)-2 is PRP (6707 digits)

List 2.b, entries with 2774 < i < 10^5
...
F(4216)+2 is prime (881 digits)

F(9110)+2 is prime (1904 digits)
F(18934)+2 is PRP (3957 digits)

F(30656)+2 is PRP (6407 digits)
F(47540)+2 is PRP (9935 digits)
F(49868)+2 is PRP (10422 digits)
F(64600)+2 is PRP (13501 digits)
F(66212)+2 is PRP (13838 digits)
F(82652)+2 is PRP (17273 digits)

He added later (Aug 18) this:

My system is too slow to double-check Simon's search up to index 191500, but I verified that his large PRPs  F(154814)+2  and  F(171106)-2  also pass my Baillie-PSW test.

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RE: Q3, I obtained the following results, the given n, and plus for F(n)+2, minus for F(n)-2, which are prime.

4216 plus

6968 minus

9110 plus

10042 minus

11720 minus

12650 minus

15604 minus

15820 minus

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Simon wrote:

Q3: These are probably-primes:

F( 6968 )-2=Prime
F( 10042 )-2=Prime
F( 11720 )-2=Prime
F( 12650 )-2=Prime
F( 15604 )-2=Prime
F( 15820 )-2=Prime
F( 171106 )-2=Prime

F( 4216 )+2=Prime
F( 9110 )+2=Prime
F( 18934 )+2=Prime
F( 154814 )+2=Prime

Searched up to F( 191500 ).

... Later (on August 18) he added:

Oups I overlooked some results reading the output,
however my missing results were found by others.
So just for completeness:

Q3: These are probably-primes:

F( 6968 )-2=Prime
F( 10042 )-2=Prime
F( 11720 )-2=Prime
F( 12650 )-2=Prime
F( 15604 )-2=Prime
F( 15820 )-2=Prime
F( 27290 )-2=Prime
F( 28960 )-2=Prime
F( 32090 )-2=Prime
F( 171106 )-2=Prime

F( 4216 )+2=Prime
F( 9110 )+2=Prime
F( 18934 )+2=Prime
F( 30656 )+2=Prime
F( 47540 )+2=Prime
F( 49868 )+2=Prime
F( 64600 )+2=Prime
F( 66212 )+2=Prime
F( 82652 )+2=Prime
F( 154814 )+2=Prime

Searched up to 191500.

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Emmanuel wrote:

Q1.  I proved my statement in my contribution to your puzzle 789.

Q3.
The next indices  m  such that  F(m) - 2  is prime are :
6968, 10042, 11720, 12650, 15604, 15820 (no more < 20000)

The next indices  m  such that  F(m) + 2  is prime are :
4216, 9110, 18934 (no more < 20000)

***

Paul wrote:

Sorry a bit late with this, here are my findings for Q3

F(1)+2 = Prime

F(2)+2 = Prime

F(3)+1 = Prime

F(4)+2 = Prime

F(5)-2 = Prime

F(5)+2 = Prime

F(6)-1 = Prime

F(7)-2 = Prime

F(8)-2 = Prime

F(8)+2 = Prime

F(10)-2 = Prime

F(14)+2 = Prime

F(20)-2 = Prime

F(22)+2 = Prime

F(40)+2 = Prime

F(76)+2 = Prime

F(88)+2 = Prime

F(100)-2 = Prime

F(106)-2 = Prime

F(158)+2 = Prime

F(196)+2 = Prime

F(230)+2 = Prime

F(236)+2 = Prime

F(260)+2 = Prime

F(508)-2 = Prime

F(586)-2 = Prime

F(788)+2 = Prime

F(800)+2 = Prime

F(938)-2 = Prime

F(1162)-2 = Prime

F(1244)-2 = Prime

F(1280)+2 = Prime

F(1444)-2 = Prime

F(1492)-2 = Prime

F(2374)+2 = Prime

F(2774)+2 = Prime

F(2884)-2 = Prime

F(3124)-2 = Prime

F(3170)-2 = Prime

F(4216)+2 = Prime

F(4564)-2 = Prime

F(6968)-2 = Prime

F(9110)+2 = Prime

F(10042)-2 = Prime

F(11720)-2 = Prime

F(12650)-2 = Prime

F(15604)-2 = Prime

F(15820)-2 = Prime

F(18934)+2 = Prime

F(27290)-2 = Prime

F(28960)-2 = Prime

F(30656)+2 = Prime

F(32090)-2 = Prime

F(47540)+2 = Prime

F(49868)+2 = Prime

F(64600)+2 = Prime

F(66212)+2 = Prime

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