## Re: composite trinomials

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• ... Yes, this submitter is mistaken in her/his proving abilities: 2^65536-2^256+1 has factors: 4933 2^262144-2^512+1 has factors: 233 Thanks for vigilance,
Message 1 of 11 , Jul 1, 2002
Paul Underwood noted:
> 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.
Yes, this submitter is mistaken in her/his
proving abilities:
2^65536-2^256+1 has factors: 4933
2^262144-2^512+1 has factors: 233
Thanks for vigilance, Paul.
David
• Note, One should not waste time on prp tests or even proof of numbers which look fishy without first trying at least a little bit of factoring. Within a few
Message 2 of 11 , Jul 1, 2002
Note,

One should not waste time on prp tests or even proof of numbers
which look "fishy" without first trying at least a little bit
of factoring. Within a few seconds (most of that was my typing),
I found this out:

C:\prime>pfgw -f -q2^65536-2^256+1
PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')

trial factoring to 5980328
2^65536-2^256+1 has factors: 4933

C:\prime>pfgw -f -q2^262144-2^512+1
PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')

trial factoring to 26465419
2^262144-2^512+1 has factors: 233

So it is very easy to see that neither of these could possibly be
prime.

Jim.
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')
>
> PRP: 2^65536-2^256+1 65000/65535
> Done.
> PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')
>
> PRP: 2^262144-2^512+1 262143/262143
> Done.
> Not even a PRP.
> Paul
• David and Jim use the -f switch. I usually forget! I tend to use sieves before OpenPFGW and so I do not use -f. :) Paul
Message 3 of 11 , Jul 1, 2002
David and Jim use the -f switch. I usually forget! I tend to use
sieves before OpenPFGW and so I do not use -f. :)
Paul
• ... Thanks--I was verysuspicious myself and had written to x37, no answer--but now none is needed. CC
Message 4 of 11 , Jul 1, 2002
On Mon, 1 Jul 2002, djbroadhurst wrote:

> Paul Underwood noted:
> > 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.
> Yes, this submitter is mistaken in her/his
> proving abilities:
> 2^65536-2^256+1 has factors: 4933
> 2^262144-2^512+1 has factors: 233
> Thanks for vigilance, Paul.
> David

Thanks--I was verysuspicious myself and had written to x37,
no answer--but now none is needed.

CC
• ... Hi Folks! Actually, the first one doesn t look too bad at all. 65536-256 = 65280. 65280 has 72 divisors, which means N-1 also has 72 cyclotomic factors,
Message 5 of 11 , Jul 2, 2002
--- paulunderwooduk <paulunderwood@...> 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.

Hi Folks!

Actually, the first one doesn't look too bad at all.

65536-256 = 65280.

65280 has 72 divisors, which means N-1 also has 72 cyclotomic factors, not
counting Aurifeullians that is. Might be worth a look. But provable or not it's
still not archivable. But a nice challenge....

Bouk.

> PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')
>
> PRP: 2^65536-2^256+1 65000/65535
> Done.
> PFGW Version 20020515.Win_Dev (Beta software, 'caveat utilitor')
>
> PRP: 2^262144-2^512+1 262143/262143
> Done.
> Not even a PRP.
> Paul
>
>
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>

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• ... Sorry, BOTH are not prp. I thought Paul only meant the largest one. Bouk. __________________________________________________ Do You Yahoo!? Sign up for SBC
Message 6 of 11 , Jul 2, 2002
--- paulunderwooduk <paulunderwood@...> 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.

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• I have a yahoo mailbox myself and sometimes I get messages from other members of the group many hours, sometimes more than a day late. 6 people had already
Message 7 of 11 , Jul 2, 2002
I have a yahoo mailbox myself and sometimes I get messages from other members
of the group many hours, sometimes more than a day late.

6 people had already discussed the composite sumbmitted numbers by x37 and I
had only received Paul's announcement. Do more people have this problem? They
are quite eager to tell me how to do something about my hairloss or shrink my
ass in twenty days but sending through messages is obviously not making them
money.

Bouk.

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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 8 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 9 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 10 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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