I was playing around with the Fibonacci sequence and the Lucas sequence, and I noticed something.
It seems that for most $n\in\mathbb{N}$ where $n|(F_n+L_n)$, $n$ is prime and is congruent to 3 or 7 modulo 10. As far as I can tell, no primes that are congruent to 1 or 9 modulo 10 fit this description.
For example, for primes ending in 3 or 7:
- $F_{13}+L_{13}=233+521=754=13\cdot58$
- $F_{17}+L_{17}=1597+3571=5168=17\cdot304$
- $F_{37}+L_{37}=24157817+54018521=78176338=37\cdot2112874$
- $F_{83}+L_{83}=99194853094755497+221806434537978679=321001287632734176=83\cdot3867485393165472$
And for good measure, here are a few primes ending in 1 or 9 (that don't divide the aforementioned sum):
- $F_{11}+L_{11}=89+199=288=2^5\cdot3^2$
- $F_{19}+L_{19}=4181+9349=13530=2\cdot3\cdot5\cdot11\cdot41$
- $F_{29}+L_{29}=514229+1149851=1664080=2^4\cdot5\cdot11\cdot31\cdot61$
Is this an actual pattern, and if so, is there a reason for it?