## Carmichael question

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• Given the following series, x^2 - x - 1 can this ever form a Carmichael Number? Note that any divisor of x^2 - x - 1 has the following properties. 1.
Message 1 of 77 , Oct 28, 2002
Given the following series, x^2 - x - 1 can this ever form a Carmichael
Number?

Note that any divisor of x^2 - x - 1 has the following properties.
1. divisor + 1 = a+b
2. a*b mod divisor = divisor - 1.

Example 1
31 + 1 = 13+19 and 13*19 MOD 31 = 30
This also means that 31 divides 13^2-13-1 and 19^2-19-1

Example 2
59+ 1 = 26+34and 26*34 MOD 59 = 58
This also means that 59 divides 26^2-26-1 and 34^2-34-1

If a Carmichael can be formed from this series what is the smallest
possible one?

Markus
• Apologies for the aborted attempt at a post. Tab followed by either space or return (too quick to know what happened) seems to be a lethal key
Message 77 of 77 , Apr 26 7:06 AM
Apologies for the aborted attempt at a post. 'Tab' followed by either
'space' or 'return' (too quick to know what happened) seems to be a
lethal key combination.

From: "mcnamara_gio" <mcnamara_gio@...>
>
> usually prime. I have calculated that 72% of a(n) so that n<500 is
> prime. 81% of a(n) is prime when n<5000. 85.6% of a(n) is prime when
> n<50000 and 88.5% of a(n) is prime when n<500000. I am going to find
> more prime terms in this sequence. What do you think about it?

2, 3, and 5 can never be factors. This boosts density by a factor of
(2/1)*(3/2)*(5/4) over arbitrary ranges. However, 7,11, 13 and 17 both
divide 2 of the p possible residues. This decreases density by a factor
of (5/6)*(9/10)*(11/12)*(15/16) over arbitrary ranges.

Looking at primes up to 10000, the density boost is almost exactly 2.75.
This is pretty feeble compared with Euler's famous trinomials.

Run this script in Pari/GP:

rnorm=1.0
rthis=1.0
forprime(p=2,10000,roots=polrootsmod(x^2+7*x-1,p)~;rnorm*=(p-1)/p;rthis*=(p-#roots)/p;print(p"
"rthis" "rnorm" "roots))
print(rthis/rnorm);

Research the Euler trinomials, and try the above script on them too, to see why
I say 2.75 is pretty feeble.

Phil

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