[QUOTE=Batalov;396961]I have a snapshot from those days. Let's see.
The numbers that disappeared are:
[CODE]7^296+3^296 153.6 250.1 - [URL="http://factordb.com/index.php?query=7%5E296%2B3%5E296"]done[/URL]
9^233+5^233 154.3 222.3 - not done in factorDB
9^251-2^251 155.9 239.5 - not done in factorDB
7^263+3^263 171.0 222.3 - not done in factorDB
9^233+8^233 182.3 222.3 - [URL="http://factordb.com/index.php?query=9%5E233%2B8%5E233"]done[/URL]
9^233-8^233 192.3 222.3 - done
7^263-3^263 214.6 222.3 - not done in factorDB
7^268+2^268 223.2 226.5 - done[/CODE][/QUOTE]

7,3,263-, 7,3,263+ are done.

I've entered Tom's two results, and now everything is sync'd in factorDB and in the reservation system.

[QUOTE=Batalov;397004]I've entered Tom's two results, and now everything is sync'd in factorDB and in the reservation system.[/QUOTE]

I just had 12 days worth of sieving destroyed by a NFS crash....


I have to restart 11,2,214+. Yech.

Hello,

There are a lot of nearly homogeneous Cunningham number in FactorDB. Eg:
(4^211+3^210)/90455802383341
(5^176-3^176-1)/4923045219

That's just to show one of each type.
The first type has different degrees, effectively multiplying one term by a small constant.
The second type has a small constant term added or subtracted.

Is it possible to build SNFS polynomials for either type? If it is possible how can I do it?

Chris

[QUOTE=chris2be8;397329]Hello,

There are a lot of nearly homogeneous Cunningham number in FactorDB. Eg:
(4^211+3^210)/90455802383341
(5^176-3^176-1)/4923045219

That's just to show one of each type.
The first type has different degrees, effectively multiplying one term by a small constant.
The second type has a small constant term added or subtracted.

Is it possible to build SNFS polynomials for either type? If it is possible how can I do it?

Chris[/QUOTE]

The first form is fairly easy. Without the small factor one would normally divide the expression by the second term, giving e.g.

a^n + b^n --> (a/b)^n + 1

This last expression is then just handled in the obvious way.

If there's a small constant multiplier of either term you can still just do the same thing. E.g.:

c*a^n + d*b^n --> c*(a/b)^n + d

This can still be handled in an obvious way.

So to take your concrete example of 4^211+3^210, we have a = 4, b = 3, c = 4, d = 1. So

4*4^210 + 3^210 --> 4*(4/3)^210 + 1

So for a quintic polynomial you would have 4x^5 + 1, with the linear polynomial being 3^42*x - 4^42.

I don't know what to do with your second example, though.

[QUOTE=jyb;397331]
I don't know what to do with your second example, though.[/QUOTE]Interesting question to which I also don't yet have an answer. Might ask around.

Paul

[QUOTE=xilman;397340]Interesting question to which I also don't yet have an answer. Might ask around.

Paul[/QUOTE]

Nothing comes to mind.

[QUOTE=jyb;397331]The first form is fairly easy. Without the small factor one would normally divide the expression by the second term, giving e.g.

a^n + b^n --> (a/b)^n + 1

This last expression is then just handled in the obvious way.

If there's a small constant multiplier of either term you can still just do the same thing. E.g.:

c*a^n + d*b^n --> c*(a/b)^n + d

This can still be handled in an obvious way.

So to take your concrete example of 4^211+3^210, we have a = 4, b = 3, c = 4, d = 1. So

4*4^210 + 3^210 --> 4*(4/3)^210 + 1

So for a quintic polynomial you would have 4x^5 + 1, with the linear polynomial being 3^42*x - 4^42.

I don't know what to do with your second example, though.[/QUOTE]

I realized that one thing I didn't address here was algebraic factors. When there is no small multiplier, there may be algebraic factors which you can divide out to give a more complicated polynomial but with much lower difficulty. E.g. for 4^210+3^210, since 210 is divisible by 15, you can play some tricks which will get the difficulty down from 127 to 68 (so low that it wouldn't even be a candidate for NFS, but you can imagine doing this with larger exponents).

With the multiplier though, finding such algebraic factors is much harder. I don't know of any algebraic factors for 4a^210 + 1, though others may be able to supply some. If the exponent were a multiple of 4 things would be different. E.g.:

4a^420 + 1 = (2a^210 - 2a^105 + 1) * (2a^210 + 2a^105 + 1)

All of which is to say: algebraic factors become more complicated when you have constant multipliers on either or both terms.

Thanks jyb, I've worked out how to build polys for a^b+-c^d by hand (at least when gcd(a,b)=1 and gcd(b,d)=1). Converting it into code should be just a SMOP. How many corner cases I'll find when I try to do it is another matter.

Algebraic factors will probably occur when the exponents share a small factor. I'll probably ignore them to begin with, then look at them when the commonest case is working.

Chris

Another 23 factors have been found since the last update. One of them was by ECM, the first for 15 months! The complete list is given below and the updated files will hit the web shortly. There are now 203 composites remaining. The tables will not be extended until it falls below 100. This is likely to happen before the end of this year at current rates of progress.

Paul

[code]
3+2 517 C211 7329012878389590484768637731440615380561352234307127409059. C153 S Batalov SNFS 2015-02-13
3+2 517 C153 58655171946966834057856107364302263591323554910607725849002847. P92 S Batalov SNFS 2015-02-13
6+5 319 C160 923767945795408630672648700962837704698509430372121747791466122767259. P91 S Batalov SNFS 2015-02-13
7+4 299 C204 3183189803421486682313612796128178272508203993846441338178551968387317465929619. P125 S Batalov SNFS 2015-02-15
7+6 263 C216 1073941292715763883388942483017613208836941311355946850919828472271139308134988697135039409157967947. P117 S Batalov SNFS 2015-02-15
7+6 299 C176 157477147053262082403867612993465484249574485422473799263291609714531383. P105 S Batalov SNFS 2015-02-16
11-4 223 C232 4461613714766346587307743231253561803840535191467479250999201. P171 S Batalov SNFS 2015-02-16
6+5 311 C241 2650554594271966793737189674575100033074067452397199626806741. C181 S Batalov SNFS 2015-02-16
6+5 311 C181 3249478033012837675159280594564071367947544896274556547562428700879659991151008291. P100 S Batalov SNFS 2015-02-16
7-3 263 C215 255315395057727481050868531668810148009004867835570694031213. P156 R Silverman SNFS 2015-02-16
12+7 227 C244 1376647037617242752227905564172736610842733025297380117354019799793620877182644535546373325832157740316054764882807. P130 S Batalov SNFS 2015-02-16
5+4 326 C224 37696913705335558299476490225688394993783296639085797077013. C165 S Batalov SNFS 2015-02-16
5+4 326 C165 4675416796708351935790872656344198998377931386872210979677749317. P102 S Batalov SNFS 2015-02-16
7+2 268 C224 41880928682672658944763414136655950967599710481106838794873. P165 J Becker SNFS 2015-02-17
7+3 263 C171 253333366089098007403559542626254658064762532109852936867506388973. P106 R Silverman SNFS 2015-02-23
9-8 233 C193 162471305534055918816266084664469400515253537568975673273763767. P131 J Becker SNFS 2015-02-26
9+8 233 C183 14667822180734640000940663045424235209450174868330046369. P128 J Becker SNFS 2015-02-28
7+3 296 C154 107074046078807903056101555301454816873464984597347424048513. P95 J Becker GNFS 2015-03-01
11+7 214 C190 55982732203366873978384729090854655904346762696518590506515366828633. P122 J Becker SNFS 2015-03-05
9-2 251 C156 5761552932141831918424906124563557274790527303200329468881305870552769. P87 J Becker SNFS 2015-03-08
7+3 293 C172 7397212251628189338610273517218859168393386670950597. C120 J Becker ECM 2015-03-08
7+3 293 C120 3541661507710128927547463511980462469526620621689. P72 J Becker GNFS 2015-03-08
9+5 233 C155 803266270279523285785189812299345682870140098329079880860996849. P92 J Becker GNFS 2015-03-19
[/code]

[QUOTE=xilman;398348] The tables will not be extended until it falls below 100. This is likely to happen before the end of this year at current rates of progress.

Paul
[/QUOTE]

Not if jyb and I are the only ones participating......

Note also:

(1) The numbers are getting bigger, so the rate will slow somewhat.
(2) I have lost some hardware.....