The low base Brent composite 43^149-1 had 2 previously known factors of 9 and 11 digits. The remaining C224 was factored as P59 * P165.

ECM to t50 by yoyo@home
SNFS sieving by RSALS
Post Processing by Carlos Pinho
[code]49^149-1

P59: 86959139755992334146951905688612930845160757695048012660011
P165: 210379001213136221541664472540640112042495029485679467795377614362692978042544981962987197932691344997299555981733468380709579937416527275299320760668948931886000961[/code]

5^1024+1024^5
 

The loveliest factorization I ever found was that of 5^1024+1024^5. It required a lot of luck and very little factoring effort.

5^1024+1024^5 = 113 * 809 * 160313 * 42387041 * 7329030409 * 425520498209 * 3024735158536046561 * P324 * P335

[QUOTE=warut;228282]The loveliest factorization I ever found was that of 5^1024+1024^5. It required a lot of luck and very little factoring effort.

5^1024+1024^5 = 113 * 809 * 160313 * 42387041 * 7329030409 * 425520498209 * 3024735158536046561 * P324 * P335[/QUOTE]

Please explain. Is there an algebraic factorization here that I am missing?
This is far beyond current capabilities otherwise.

[quote=R.D. Silverman;228286]Please explain. Is there an algebraic factorization here that I am missing?
This is far beyond current capabilities otherwise.[/quote]
Yes, it admits an algebraic factorization because
5^1024 + 1024^5 = (5^256)^4 + 4*(2^12)^4.

[QUOTE=warut;228314]Yes, it admits an algebraic factorization because
5^1024 + 1024^5 = (5^256)^4 + 4*(2^12)^4.[/QUOTE]

Yes. Nice result.

643945470136695841369 | M52272617

Found with P-1 where k = 2^2 * 3 * 17 * 31 * 499 * 709 * 2753

Man, that is smooth.

Nothing too unusual about a p49 these days but this one was found by gmp-ecm using a B1 value of 1M, which is optimal for factors 14 digits smaller.

[code][2010-09-07 11:15:27 GMT] GC_5_691_C421: probable factor returned by pcl@maat (maat4)! Factor=2341673831256161998097409059802772958624071425517 Method=ECM B1=1000000 Sigma=1067295280[/code]GC_5_691 is 691*5^691+1

Paul

Top 10?
 

Why hasn't Blair Kelly's P64 factor of L1956 shown up on the
ECM Top 10 page?

5748 index 1368:
[code]
Thu Sep 30 17:37:00 2010 prp69 factor: 674788222835164103353823038231553835542017738166824174868830262813749
Thu Sep 30 17:37:00 2010 prp86 factor: 28250192298752567946214115957812098958537842561773991296565797999910836802368478096257
[/code]

The Brent composite 541^89-1 had 3 previously known factors of 3, 6, and 11 digits. The remaining C223 has been factored as P89 * P135.

ECM to t50 by yoyo@home
SNFS sieving by RSALS
Post Processing by ZetaFlux
[code]541^89-1

P89: 22258230882835545380142293457243394259854811189356144633053908389611484579705336039818299
P135: 156308318050822004478554684741294957177596514375932813652522859001742185442920716992906314508709642460414410563253635690990135902349719[/code]

a factor of 220 digits...
 

[FONT=Arial Narrow]> echo "(10^9270-1)/3^4/7" | ecm 5e4
Input number is (10^9270-1)/3^4/7 (9268 digits)
Using B1=50000, B2=15446350, polynomial x^2, sigma=3682263502
Step 1 took 99069ms
********** Factor found in step 1: 9573726981993820200551709701677247363792903115583020462044881149461982541836124423370428786937847788404630054782651343281856841362377745746844353951111544591359795066465415084994921961147188454685295861838434800628490443
Found composite factor of 220 digits: 9573726981993820200551709701677247363792903115583020462044881149461982541836124423370428786937847788404630054782651343281856841362377745746844353951111544591359795066465415084994921961147188454685295861838434800628490443
Composite cofactor ((10^9270-1)/3^4/7)/9573726981993820200551709701677247363792903115583020462044881149461982541836124423370428786937847788404630054782651343281856841362377745746844353951111544591359795066465415084994921961147188454685295861838434800628490443 has 9048 digits
[/FONT]
[FONT=Arial Narrow]:wblipp:[/FONT]
[FONT=Arial Narrow]:rolleyes:[/FONT]