## Re: Carmichael question

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• Congrats on a neat method, Richard. ... How did you rediscover Jack s 675557402^2-675557402-1 = 456377802721432201 = 29*211*281*22669*11708611 where 22669-1 =
Message 1 of 77 , Nov 3, 2002
Congrats on a neat method, Richard.

However one things puzzles me:

> it trots out the Carmichaels
> everyone else has found,
> in a couple of minutes on a P3-800.

How did you rediscover Jack's

675557402^2-675557402-1 =
456377802721432201 =
29*211*281*22669*11708611

where

22669-1 = 2^2*3*1889
11708611-1 = 2*3*5*23*71*239

are not so smooth?

The seeds {211,281} are too small
to generate a result from the
residue class of n=675557402 modulo
m=211*281*lcm(210,280)=49804440

With n/m>13, you would have needed 14 attempts at
n=k*m+r
on _each_ of your 128 CRT residue classes
no so?

Note that Jack's Carmichael has only 18 digits,
whereas your 3 new finds have 23 digits.

I suspect that there are quite a
few n^2-n-1 Carmichaels below 23 digits
that cannot be found by a double
loop of residue classes from seeds {p,q}
with both p-1 and q-1
as smooth as your are hoping.

But still, yours is a really neat way to
find a subset of the targets!

Here's another 23-digit find:

151245190802^2-151245190802-1 =
22875107740582140212401 =
271*1151*3221*6841*36541*91081

I think this is #10 and the largest yet?

Best regards

David
• 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, 2005
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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