## Re: composite trinomials

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• ... I should watch my grammar! If you want a challange, dislodge some of my gigantic PRP trinomials at Henri s site:
Message 1 of 11 , Jul 2, 2002
Bouk wrote:
> --- I wrote:
> > Hi,
> > I have just seen these:
> > 9999 2^65536-2^256+1 19729 x37 02 prime #0207
> > 9999 2^262144-2^512+1 78914 x37 02 #0207
> > which look improbable and a BLS no-hoper.
> > PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')
>
> Sorry, BOTH are not prp. I thought Paul only meant the largest one.
>
> Bouk.
I should watch my grammar!
If you want a challange, dislodge some of my gigantic PRP trinomials
at Henri's site:
If you can't do that I can generate a challanging one if you like!
Paul
• ... Actually I did browse them for proofs. One or two could be proven with a large ECM effort. This one is a good very good one to try: 2^64695-2^15-1 with
Message 2 of 11 , Jul 3, 2002
> If you want a challange, dislodge some of my gigantic PRP trinomials
> at Henri's site:

Actually I did browse them for proofs. One or two could be proven with a large
ECM effort.

This one is a good very good one to try:

2^64695-2^15-1 with 19476 digits.

N+1 = 2^15*(2^64680-1)

(2^64680-1) has 96 cyclotomic divisors.

T 64680={ 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 20, 21, 22, 24, 28, 30,
33, 35, 40, 42, 44, 49, 55, 56, 60, 66, 70, 77, 84, 88, 98, 105, 110, 120, 132,
140, 147, 154, 165, 168, 196, 210, 220, 231, 245, 264, 280, 294, 308, 330, 385,
392, 420, 440, 462, 490, 539, 588, 616, 660, 735, 770, 840, 924, 980, 1078,
1155, 1176, 1320, 1470, 1540, 1617, 1848, 1960, 2156, 2310, 2695, 2940, 3080,
3234, 4312, 4620, 5390, 5880, 6468, 8085, 9240, 10780, 12936, 16170, 21560,
32340, 64680 } [96]

So: (2^64680-1) =
phi(1,2)*phi(2,2)*phi(3,2)*phi(4,2)*.....*phi(21560,2)*phi(32340,2)*phi(64680,2)

There are aurifeuillian factors (L and M) as well when for phi(n,2) n=4*k and
k=odd.

L=2^h-2^k+1, M=2^h+2^k+1, h=2k-1. Take gcd's with phi(4*k,2) and L or M.

E.g. phi(2156,2) can be divided in a L and M part as 2156 = 4*539

L: 10781.81929.90317512080398683509507180285854441.P83
M:
2136469147429.111206916097779728932051224808777.1297662995123479965752936319854262257.P46

There is great deal of work already done in the cunninghamproject. I would
expect factorization to be more than 25% done. If one reaches 30% David
Broadhurst can do his KP magic.

Bouk de Water.

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• And for die-hards: 2^118843-2^43+1 with 35776 digits. N-1: 2^43*(2^118800-1) T 118800={ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30,
Message 3 of 11 , Jul 3, 2002
And for die-hards:

2^118843-2^43+1 with 35776 digits.

N-1: 2^43*(2^118800-1)

T 118800={ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27,
30, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 66, 72, 75, 80, 88, 90, 99, 100,
108, 110, 120, 132, 135, 144, 150, 165, 176, 180, 198, 200, 216, 220, 225, 240,
264, 270, 275, 297, 300, 330, 360, 396, 400, 432, 440, 450, 495, 528, 540, 550,
594, 600, 660, 675, 720, 792, 825, 880, 900, 990, 1080, 1100, 1188, 1200, 1320,
1350, 1485, 1584, 1650, 1800, 1980, 2160, 2200, 2376, 2475, 2640, 2700, 2970,
3300, 3600, 3960, 4400, 4752, 4950, 5400, 5940, 6600, 7425, 7920, 9900, 10800,
11880, 13200, 14850, 19800, 23760, 29700, 39600, 59400, 118800 } [120]

But that one is extremely hard. Don't try this at home, folks!

Bouk.

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