Raphie
May10-11, 09:27 PM
Thought to pass this "somewhat" random observation along for any out there who might suspect a relationship between the Golden Ratio, the distribution of the primes and lattices, not to mention partition numbers...
RELATED PROGRESSION
A134696 Numbers n such that k*Lucas(n) + 1 is prime.
1, 2, 3, 4, 6, 10, 42, 48, 56, 66, 70, 86, 108, 126, 134, 214, 248, 459, 1479, 1722
http://oeis.org/A134696
For L_n a Lucas Number, p_n a prime number and par(n) a partition number:
(known) Solutions:
01*L_1 + 1 = (01 * 01) + 1 = 0002 = p_001 = p_(L_01) = p_(L_(L_1) = 01*L_(L_1)
02*L_2 + 1 = (02 * 03) + 1 = 0007 = p_004 = p_(L_03) = p_(L_(L_2) = 02*L_(L_0)
42*L_7 + 1 = (42 * 29) + 1 = 1219 = p_199 = p_(L_11) = p_(L_(L_5) = 42*L_(L_4)
And any others?
Interestingly, for n > 1, where K_n is a (Lattice) Kissing Number, then...
02*L_2 = 02 * 03 = (K_2*L_2)/3 = (006*03)/3 = 0006
42*L_7 = 42 * 29 = (K_7*L_7)/3 = (126*29)/3 = 1218
Delta (6, 1218) = 1212 = par(13)*totient (13) = 12*101
6 = par(3)*totient (3) = 3*2
Sigma (6, 1218) = 1218 + 6 = 1224 == 10^2*K_3 + 10^0*K_4 for K_3 = 12 and K_4 = 24
Kind of "nifty" also since 42^2(base 10) = 24024(base 5) and 1,1,2,42 and 24024 are the first five "moments" of the Riemann Zeta Function which Marcus du Sautoy writes about in his book "The Musics of the Primes."
1, 2 and 42, incidentally, all follow the form x^2 + x (twice a Triangular Number for n in N):
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(1/phi)^2 + (1/phi) = (phi - 1)^2 + (phi - 1)
1^2 + 1 = (2 - 1)^2 + (2 - 1)
6^2 + 6 = (7 - 1)^2 + (7 - 1)
And 6 = (3*2) = K_2 and 126 = (3*42) = K_7 are also solutions (the 5th and 14th) to n*Lucas_n + 1 is prime. So too, 3 = (3*1), which is the 3rd Solution.
003*004 + 1 = 000013 = p_00006
006*011 + 1 = 000067 = p_00019
126*843 + 1 = 106219 = p_10124
Oh, and of course, 1, 2, and 42 are all also partition numbers...
01 = par(00)
02 = par(02)
42 = par(10)
... which, coincidentally, happens to circle back rather nicely to the original observation...
p_(L_(00 + 1)) = par(00)*L_(par_1) + 1 = p_(L_(L_1) = (01 * 01) + 1 = 0002
p_(L_(02 + 1)) = par(02)*L_(par_2) + 1 = p_(L_(L_2) = (02 * 03) + 1 = 0007
p_(L_(10 + 1)) = par(10)*L_(par_5) + 1 = p_(L_(L_5) = (42 * 29) + 1 = 1219
Best,
RF
Source of Observation: An OEIS Search for 1,2,3,4,6, & 10. These numbers are all related to n-fold rotational symmetry. 1,2,3,4 & 6 are the only integer > 0 solutions to 2*cos (2*pi/n) is in N, while 10, when inserted into the same equation = phi, the Golden Ratio, associated with aperiodic crystal growth (i.e quasicrystals).
2*cos (2*pi/1) = 1
2*cos (2*pi/2) = -2
2*cos (2*pi/3) = -1
2*cos (2*pi/4) = 0
2*cos (2*pi/6) = 1
2*cos (2*pi/10) = Golden Ratio
(Sum, if viewed as a system = 1/phi = -1 + phi)
RELATED LINKS
The Crystallographic Restriction Theorem
http://en.wikipedia.org/wiki/Crystallographic_restriction_theorem
Quasicrystals
http://en.wikipedia.org/wiki/Quasicrystal
RELATED PROGRESSION
A134696 Numbers n such that k*Lucas(n) + 1 is prime.
1, 2, 3, 4, 6, 10, 42, 48, 56, 66, 70, 86, 108, 126, 134, 214, 248, 459, 1479, 1722
http://oeis.org/A134696
For L_n a Lucas Number, p_n a prime number and par(n) a partition number:
(known) Solutions:
01*L_1 + 1 = (01 * 01) + 1 = 0002 = p_001 = p_(L_01) = p_(L_(L_1) = 01*L_(L_1)
02*L_2 + 1 = (02 * 03) + 1 = 0007 = p_004 = p_(L_03) = p_(L_(L_2) = 02*L_(L_0)
42*L_7 + 1 = (42 * 29) + 1 = 1219 = p_199 = p_(L_11) = p_(L_(L_5) = 42*L_(L_4)
And any others?
Interestingly, for n > 1, where K_n is a (Lattice) Kissing Number, then...
02*L_2 = 02 * 03 = (K_2*L_2)/3 = (006*03)/3 = 0006
42*L_7 = 42 * 29 = (K_7*L_7)/3 = (126*29)/3 = 1218
Delta (6, 1218) = 1212 = par(13)*totient (13) = 12*101
6 = par(3)*totient (3) = 3*2
Sigma (6, 1218) = 1218 + 6 = 1224 == 10^2*K_3 + 10^0*K_4 for K_3 = 12 and K_4 = 24
Kind of "nifty" also since 42^2(base 10) = 24024(base 5) and 1,1,2,42 and 24024 are the first five "moments" of the Riemann Zeta Function which Marcus du Sautoy writes about in his book "The Musics of the Primes."
1, 2 and 42, incidentally, all follow the form x^2 + x (twice a Triangular Number for n in N):
-----------------------------------------------------------------------------------------------------
(1/phi)^2 + (1/phi) = (phi - 1)^2 + (phi - 1)
1^2 + 1 = (2 - 1)^2 + (2 - 1)
6^2 + 6 = (7 - 1)^2 + (7 - 1)
And 6 = (3*2) = K_2 and 126 = (3*42) = K_7 are also solutions (the 5th and 14th) to n*Lucas_n + 1 is prime. So too, 3 = (3*1), which is the 3rd Solution.
003*004 + 1 = 000013 = p_00006
006*011 + 1 = 000067 = p_00019
126*843 + 1 = 106219 = p_10124
Oh, and of course, 1, 2, and 42 are all also partition numbers...
01 = par(00)
02 = par(02)
42 = par(10)
... which, coincidentally, happens to circle back rather nicely to the original observation...
p_(L_(00 + 1)) = par(00)*L_(par_1) + 1 = p_(L_(L_1) = (01 * 01) + 1 = 0002
p_(L_(02 + 1)) = par(02)*L_(par_2) + 1 = p_(L_(L_2) = (02 * 03) + 1 = 0007
p_(L_(10 + 1)) = par(10)*L_(par_5) + 1 = p_(L_(L_5) = (42 * 29) + 1 = 1219
Best,
RF
Source of Observation: An OEIS Search for 1,2,3,4,6, & 10. These numbers are all related to n-fold rotational symmetry. 1,2,3,4 & 6 are the only integer > 0 solutions to 2*cos (2*pi/n) is in N, while 10, when inserted into the same equation = phi, the Golden Ratio, associated with aperiodic crystal growth (i.e quasicrystals).
2*cos (2*pi/1) = 1
2*cos (2*pi/2) = -2
2*cos (2*pi/3) = -1
2*cos (2*pi/4) = 0
2*cos (2*pi/6) = 1
2*cos (2*pi/10) = Golden Ratio
(Sum, if viewed as a system = 1/phi = -1 + phi)
RELATED LINKS
The Crystallographic Restriction Theorem
http://en.wikipedia.org/wiki/Crystallographic_restriction_theorem
Quasicrystals
http://en.wikipedia.org/wiki/Quasicrystal