The Lucas Sequence

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349

 

The Lucas numbers unfortunately aren’t as famous as they should be but many would still recognize it’s rule from a more popular sequence called the Fibonacci sequence. Same rule as the Fibonacci sequence only the start is switched from 1, 1, to 2, 1.

36087311_658422854494345_2570246070743334912_n.jpg

Any two numbers that you apply this rule to the ratio of the numbers will get you a closer and closer approximation to Phi Φ. 

Lucas Numbers in blue

 

  1. 1
  2. 3
  3. 4
  4.  — 7  
  5.  — 11   
  6. 18   
  7. —  29
  8. 47
  9. 76
  10. 123
  11. 199
  12. 322
  13. 521
  14. 843
  15. 1364
  16. 2207
  17. 3571
  18. 5778
  19. 9349

 

Lucas Number and φ

 

2/1= 1

3/2= 1.5

4/3= 1.3333 —

7/4= 1.75

11/7= 1.57142857

18/11= 1.63636364

29/18= 1.611111 — 

47/29= 1.62068

76/47= 1.61702128

123/76= 1.61842105

199/123= 1.61788618

322/199= 1.61809045

 

Every power of Phi rounded to the nearest whole number will give you a Lucas number..

 

φ² = 2.61803 —

φ³= 4.23606 — 4

φ^4= 6.85440 — 7

φ^5= 11.09016 — 11

φ^6= 17.94427 — 18

φ^7= 29.03444 — 29

φ^8= 46.97871— 47

 

 

 

The first 20 Primes in the Lucas Numbers

 

— 6 of the first 9 Lucas numbers are prime.

— 10 primes are found within the first 20 Lucas numbers.

That’s a lot of primes close together!

 

Primes in purple

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322,

521, 843, 1364, 2207, 3571, 5778, 9349

 

3571 is the 500th prime number and the 70th from 3001-4000.

 

The Fibonacci sequence gives you this pattern…

ODD ODD EVEN ODD ODD EVEN

 

The Lucas numbers gives you this pattern…

EVEN ODD ODD EVEN ODD ODD

Primes in the Fibonacci sequence

 

— 3 primes within the first 9 numbers.

— 6 primes within the first 20 numbers.

 

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,

377, 610, 987, 1597, 2584, 4181, 6765

 

 

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