
Perrin Sequence - ColinWright
http://www.cut-the-knot.org/arithmetic/combinatorics/PerrinSequence.shtml
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ColinWright
It was once presented to me as a fact, and they asked for a proof, that n|k(n)
if and only if n is prime. I immediately said that it couldn't possibly be
true, so I was challenged to find a counter-example.

This was in the days before languages like Python, with arbitrary precision
integers. I used C to find the first place it failed, and indeed, all counter-
examples less than 2^32.

There aren't many, and they have interesting things in common. It's a
fascinating mathematical and programmatic playground.

~~~
Chinjut
Impressive! Out of curiosity, did you have good reason to immediately suspect
it couldn't possibly be true? According to Wikipedia, this question had been
considered considered by Perrin himself (way back in 1899), but was only first
resolved by Adam and Shanks in 1982.

~~~
peterderivaz
I don't know Colin's logic, but my personal reasoning would be:

1\. If this were true then it gives a simple, efficient, deterministic method
to test for primes. (You can compute P_n modulo n efficiently using matrix
exponentiation.)

2\. If there was such a method I am sure I would have heard of it before - and
people wouldn't bother using Miller-Rabin or the complicated deterministic
primality method

3\. Therefore there must be a flaw...

~~~
ColinWright
Actually, there is a polynomial algorithm for testing primality, the AKS
primality test.

[http://en.wikipedia.org/wiki/AKS_primality_test](http://en.wikipedia.org/wiki/AKS_primality_test)

In essence, my reasoning was simply that primes don't behave like this. Any
connection with addition-type stuff is spurious and doesn't go on forever.

Mind you, when I'd searched up to 10^5 and still hadn't found a counter-
example I was starting to doubt my intuition. The first counter-example is
when the sequence predicts that 521^2 is prime.

In fact, if p is prime then p divides k(p). I find that non-obvious, but do
follow and believe the proof. Even so, I don't know it well enough to feel
enlightened by it.

The work continues.

