Fibonacci and Twin Primes

[T. D. Noe]

Due to Catalan's identity for r=2:
http://mathworld.wolfram.com/CatalansIdentity.html
it is clear that every pair of prime Fibonacci numbers whose indices
are twin primes deliver an element of A140362.

On the other hand, if a product of two primes p*q belongs to A140362 then
(*)  1 + p^2 + q^2 = k*p*q for some integer k>1.

If k is even, say, k=2*m, then (*) is equivalent to:
(p - m*q)^2 - (m^2 - 1)*q^2 = -1.
This Pell-like equation does not have solutions since sqrt(m^2 - 1) is
represented by the following continued fraction:
sqrt(m^2 - 1) = [m-1; ( 1, 2*(m-1) ) ] where ( ) enclose the period.
See http://mathworld.wolfram.com/PellEquation.html for details.

For k=3, the integer solution to (*) is p=F(2m-1), q=F(2m+1), or vice versa.

For odd k>=5, (*) can be rewritten as
(**) (2*p + k*q)^2 - (k^2 - 4)*q^2 = -4
where sqrt(k^2-4)>|-4|=4. Therefore, this equation has a solution only
if -4 can be expressed as
-4 = p(n)^2 - (k^2 - 4)*q(n)^2,
where p(n)/q(n) is a convergent fraction to sqrt(k^2-4), and n does
not exceed the length of the period of the continued fraction.

Note that for k = 2t + 1 >= 5 (that is t>=2), the continued fraction
of sqrt((2t+1)^2 - 4) is:
sqrt((2t+1)^2 - 4) = [2t; ( 1, t-1, 2, t-1, 1, 4t ) ],
giving the sequence of convergents:
p(n): 2t, 2t+1, 2t^2+t-1, 4t^2+4t-1, 4t^3+2t^2-4t, 4t^3+6t^2-1,
16t^4+28t^3+2t^2-8t, ...
q(n): 1, 1, t, 2t+1, 2t^2-1, 2t^2+2t, 8t^3+10t^2-1, ...
The corresponding values of p(n)^2 - (k^2 - 4)*q(n)^2 are:
-4t + 3, 4, -2t + 1, 4, -4t + 3, 1, -4t + 3
and none of them equals -4.

Therefore, all terms of A140362 are of the form F(r-1)*F(r+1) where
r-1, r+1, F(r-1), F(r+1) are primes. Correspondingly, the first terms
of A140362 are:
F(3)*F(5), F(5)*F(7), F(11)*F(13), F(431)*F(433), F(569)*F(571)

Regards,
Max

On Fri, Jul 25, 2008 at 12:26 PM, T. D. Noe <noe at sspectra.com> wrote:
> Sequence A140362 (Integers n which are the product of two distinct primes
> and which divide the sum of the squares of the divisors of n) is currently
> has only three terms: 10, 65, 20737.
>
> For a while I was sure that the sequence was complete. Then I noticed that
> the terms could be written as Fib(3)*Fib(5), Fib(5)*Fib(7), and
> Fib(11)*Fib(13); that is, the product of prime Fibonacci numbers whose
> indices are twin primes!
>
> Using A001605, two additional terms should be Fib(431)*Fib(433) and
> Fib(569)*Fib(571):
>
> 735108038169226697610336266421235332619480119704052339198145857119174445190576122619635288017445230931072695163057441061367078715257112965183856285090884294459307720873196474208257
>
> and
>
> 3523220957390444959595279062040480245884253791540018496569589759612684974224639027640287843213615446328687904372189751725183659047971600027111855728553282782938238390010064604217978755993551604318057918269182928456761611403668577116737601
>
> So the question is: are there other solutions?
>
> Tony
>




One of the new web pages will soon be ready for testing.

This is the page to use if you are just adding something
(e.g. a comment, reference, program, formula, etc.)

Until then, please hold off on sending such comments!

Thanks!

Neil

.,




While reading Pieter Morees treatise on Witt transforms, it
is interesting to see that many of the associated sequences are
(as expected) in the OEIS.

The 1st Witt transform of a series is the series.

The 2nd Witt transform of A000012 gives A110654
The 3rd Witt transform of A000012 gives A001840
The 4th Witt transform of A000012 gives A006918
The 5th Witt transform of A000012 gives A051170
The 6th Witt transform of A000012 gives zero followed by A011796
The 7th Witt transform of A000012 gives A011797 with only 1 leading zero
The 8th Witt transform of A000012 gives zero followed by A031164

The 2nd Witt transform of A000027 gives A006584
The 3rd Witt transform of A000027 gives 0,0,0,0,2,7,18,42,84,153,264,429,666,1001,1456,2061,2856,3876,5166,6783,8778,...

The 2nd Witt transform of A000045 gives two zeros followed by A089089
The 3rd Witt transform of A000045 gives three zeros followed by A089116
The 4th Witt transform of A000045 gives four zeros followed by A089117

The 2nd Witt transform of A000217 gives three zeros followed by A032092
The 3rd Witt transform of A000217 gives 0,0,0,0,3,15,54,165,429,999,2145,4290,8100,14586,25194,41985,67830,106590,..

The 2nd Witt transform of A000292 gives three zeros followed by A032094
The 3rd Witt transform of A000292 gives 0,0,0,0,4,26,120,455,1456,4122,10608,25194,55980,117572,235144,450681,...

The 2nd Witt transform of A033999 gives essentially a signed versio of A004524
The 3rd Witt transform of A033999 gives (-1)^n*A001840(n).

The 2nd Witt transform of A040000 gives zero followed by A042964

The 2nd Witt transform of A000108 gives zero followed by A000150
The 3rd Witt transform of A000108 gives A050181
The 4th Witt transform of A000108 gives A050182
The 5th Witt transform of A000108 gives A050183

The 2nd Witt transform of A000079 gives zero followed by A124720
The 3rd Witt transform of A000079 gives zero followed by A124721
The 4th Witt transform of A000079 gives zero followed by A124722

The 2nd Witt transform of A000290 gives 0,0,0,4,15,52,125,280,538,984,1654,2684,4129,6188,8939,12656,17444,23664,..

The 2nd Witt transform of A000244 gives zero followed by A124810
The 3rd Witt transform of A000244 gives zero followed by A124811

The 2nd Witt transform of A004526 gives 5 zeros followed by A008804

Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
Pieter Moree, <a href="http://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO].

Richard Mathar




> From seqfan-owner at ext.jussieu.fr  Sat Jul  5 00:08:45 2008
> Date: Sat, 05 Jul 2008 00:07:04 +0430
> From: Artur <grafix at csl.pl>
> Reply-To: grafix at csl.pl
> To: Richard Mathar <mathar at strw.leidenuniv.nl>
> CC: seqfan at ext.jussieu.fr
> Subject: Re: Q: prime number congruences and quadratic forms: A141778=A038977?
>
> Also
> A141158 <http://www.research.att.com/%7Enjas/sequences/A141158>=A141130
> <http://www.research.att.com/%7Enjas/sequences/A141130>=A038872
> <http://www.research.att.com/%7Enjas/sequences/A038872>
> A141131 <http://www.research.att.com/%7Enjas/sequences/A141131>=A038873
> <http://www.research.att.com/%7Enjas/sequences/A038873>=A049594
> <http://www.research.att.com/%7Enjas/sequences/A049594>=A049590
> <http://www.research.att.com/%7Enjas/sequences/A049590> (titles
> different sequences that same)
> A141186 <http://www.research.att.com/%7Enjas/sequences/A141186>=A141122
> <http://www.research.att.com/%7Enjas/sequences/A141122>=A068228
> <http://www.research.att.com/%7Enjas/sequences/A068228>
> A141123? <http://www.research.att.com/%7Enjas/sequences/A141123>=A141187
> <http://www.research.att.com/%7Enjas/sequences/A141187>
> A141132 <http://www.research.att.com/%7Enjas/sequences/A141132>=A141188
> <http://www.research.att.com/%7Enjas/sequences/A141188>=A038883
> <http://www.research.att.com/%7Enjas/sequences/A038883>
> A141133 <http://www.research.att.com/%7Enjas/sequences/A141133>=A038889
> <http://www.research.att.com/%7Enjas/sequences/A038889>
> A141130 <http://www.research.att.com/%7Enjas/sequences/A141130>=A038872
> <http://www.research.att.com/%7Enjas/sequences/A038872>=A141158 also
> <http://www.research.att.com/%7Enjas/sequences/A141158>A123976 differ
> only a(1) <http://www.research.att.com/%7Enjas/sequences/A123976>
> A141159 <http://www.research.att.com/%7Enjas/sequences/A141159> also
> title that same as A139492
> <http://www.research.att.com/%7Enjas/sequences/A139492>
> A141170 <http://www.research.att.com/%7Enjas/sequences/A141170>=A106950
> <http://www.research.att.com/%7Enjas/sequences/A106950>
> A141172 <http://www.research.att.com/%7Enjas/sequences/A141172>?=A107134
> <http://www.research.att.com/%7Enjas/sequences/A107134>
> A141174 <http://www.research.att.com/%7Enjas/sequences/A141174>=A007519
> <http://www.research.att.com/%7Enjas/sequences/A007519>
> A141175 <http://www.research.att.com/%7Enjas/sequences/A141175>=A007522
> <http://www.research.att.com/%7Enjas/sequences/A007522>
> A141177 <http://www.research.att.com/%7Enjas/sequences/A141177>?=A107013
> <http://www.research.att.com/%7Enjas/sequences/A107013>
> A141178 <http://www.research.att.com/%7Enjas/sequences/A141178>=A038913
> <http://www.research.att.com/%7Enjas/sequences/A038913>
> Best wishes
> Artur


In some further cases we can show that these quadratic forms produce
primes which are in some older sequences defined via linear forms. Again, this
is only a "halfway" proof that these might be the same.

In detail:

If we compute the quadratic form in A141376 modulo 24 by scanning all
x-values from 0 to 23 and all y-values from 0 to 23 we get the
remainders 0,8,12,15,20,23. The only value with gcd(,.24)=1 is 23
(or -1 mod 24), which shows that A141376 is a subsequence of A134517.

If we compute the quadratic form in A141187 modulo 12 by scanning all
x-values from 0 to 11 and all y-values from 0 to 11 we get the
remainders 0,3,8,11. The only values with gcd(,12)=1 is 11, which
are also in A068231.

If we compute the quadratic form in A141186 modulo 12 by scanning all
x-values from 0 to 11 and all y-values from 0 to 11 we get the
remainders 0,1,4,9. The only values with gcd(,12)=1 is 1, which

If we compute the quadratic form in A141175 modulo 8 by scanning all
x-values from 0 to 7 and all y-values from 0 to 7 we get the
remainders 0,4,7. The only values with gcd(,8)=1 is 7, which

If we compute the quadratic form in A141174 modulo 8 by scanning all
x-values from 0 to 7 and all y-values from 0 to 7 we get the
remainders 0,1,4. The only values with gcd(,8)=1 is 1, which

A141163 is a duplicate of A139602 by definition.
A141161 is a duplicate of A139599 by definition.

A141162 is just another duplicate of A139599: note that one can
-Q(4x^2+6xy-7y^2), the same quadratic form up to a sign.
The sign doesn't matter because with the positive discriminant we have
an indefinite form anyway from which only the positive primes emerge.  

If we compute the quadratic form in A141158 modulo 5 by scanning all
x-values from 0 to 4 and all y-values from 0 to 4 we get the
remainders 0,1,4, which

If we compute the quadratic form in A141131 modulo 8 by scanning all
x-values from 0 to 7 and all y-values from 0 to 7 we get the
remainders 0,1,2,4,6,7, which
of the form 4 mod 8 or 6 mod 8.

If we compute the quadratic form in A141122 modulo 12 by scanning all
x-values from 0 to 11 and all y-values from 0 to 11 we get the
remainders 0,1,4,6,9,10 which
of the form {4,6,9,10} mod 12.

If we compute the quadratic form in A139492 modulo 6 by scanning all
x-values from 0 to 5 and all y-values from 0 to 5 we get the
remainders 0,1,3,4 which
of the form {3,4} mod 6.

The same rendered into internal OEIS format:

%I A141376
%S A141376 23,47,71,167,191,239,263,311,359,383,431,479,503,599,647,719,743,839,
%T A141376 863,887,911,983
%N A141376 Primes of the form -x^2+8*x*y+8*y^2 (as well as of the form 15*x^2+24*x*y+8*y^2).
%C A141376 Values of the quadr. form are {0,8,12,15,20,23} mod 24, so this is a subset of A134517. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A141187
%S A141187 3,11,23,47,59,71,83,107,131,167,179,191,227,239,251,263,311,347,359,
%T A141187 383,419,431,443,467,479,491,503,563,587,599,647,659,683,719,743,827,
%U A141187 839,863,887,911,947,971,983
%N A141187 Primes of the form -x^2+6*x*y+3*y^2 (as well as of the form 8*x^2+12*x*y+3*y^2).
%C A141187 Values of the quadr. form are {0,3,8,11} mod 12, so all values with the exception of 3 are also in A068231. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A141186
%S A141186 13,37,61,73,97,109,157,181,193,229,241,277,313,337,349,373,397,409,421,
%T A141186 433,457,541,577,601,613,661,673,709,733,757,769,829,853,877,937,997
%N A141186 Primes of the form x^2+6*x*y-3*y^2 (as well as of the form 4*x^2+8*x*y+y^2).
%C A141186 Values of the quadr. form are {0,1,4,9} mod 12, so this is a subset of A068228. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A141175
%S A141175 7,23,31,47,71,79,103,127,151,167,191,199,223,239,263,271,311,359,367,
%T A141175 383,431,439,463,479,487,503,599,607,631,647,719,727,743,751,823,839,
%U A141175 863,887,911,919,967,983,991
%N A141175 Primes of the form -x^2+4*x*y+4*y^2 (as well as of the form 7*x^2+12*x*y+4*y^2).
%C A141175 Values of the quadr. form are {0,4,7} mod 8, so this is a subset of A007522. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A141174
%S A141174 17,41,73,89,97,113,137,193,233,241,257,281,313,337,353,401,409,433,449,
%T A141174 457,521,569,577,593,601,617,641,673,761,769,809,857,881,929,937,953,
%U A141174 977
%N A141174 Primes of the form x^2+4*x*y-4*y^2 (as well as of the form x^2+6*x*y+y^2).
%C A141174 Values of the quadr. form are {0,1,4} mod 8, so this is a subset of A007519. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A141158
%S A141158 5,11,19,29,31,41,59,61,71,79,89,101,109,131,139,149,151,179,181,191,
%T A141158 199,211,229,239,241,251,269,271,281,311,331,349,359,379,389,401,409,
%U A141158 419,421,431,439,449,461,479,491,499,509,521,541,569,571,599,601,619
%N A141158 Primes of the form x^2+4*x*y-y^2.
%C A141158 Values of the quadr. form are {0,1,4} mod 5, so this is a subset of A038872. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A141131
%S A141131 2,7,17,23,31,41,47,71,73,79,89,97,103,113,127,137,151,167,191,193,199,
%T A141131 223,233,239,241,257,263,271,281,311,313,337,353,359,367,383,401,409,
%U A141131 431,433,439,449,457,463,479,487,503,521,569,577,593,599,601,607,617
%N A141131 Primes of the form x^2+2*x*y-y^2.
%C A141131 Values of the quadr. form are {0,1,2,4,6,7} mod 8, so this is a subset of A038873. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A141122
%S A141122 13,37,61,73,97,109,157,181,193,229,241,277,313,337,349,373,397,409,421,
%T A141122 433,457,541,577,601,613,661,673,709,733,757,769,829,853,877,937,997
%N A141122 Primes of the form x^2+2*x*y-2*y^2 (as well as of the form x^2+4*x*y+y^2).
%C A141122 Values of the quadr. form are {0,1,4,6,9,10} mod 12, so this is a subset of A068228. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008

%I A139492
%S A139492 7,37,43,67,79,109,127,151,163,193,211,277,331,337,373,379,421,457,463,
%T A139492 487,499,541,547,571,613,631,673,709,739,751,757,823,877,883,907,919,
%U A139492 967,991,1009,1033,1051,1087,1093,1117,1129,1171,1201,1213,1297,1303
%N A139492 Primes of the form x^2 + 5x*y + y^2 for x and y nonnegative.
%C A139492 Values of the quadr. form are {0,1,3,4} mod 6, so this is a subset of A002476. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 30 2008