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Questa pagina contiene i risultati ottenuti negli anni durante la mia attività di ricerca nell'ambito della Teoria dei Numeri.
Tutte le informazioni inerenti agli avanzamenti, personali e del Team Italia (fondato dal sottoscritto e da Guido Lorenzini), relativi ai progetti distribuiti come GIMPS, FermatSearch e Operation Billion Digits trovano posto nelle relative pagine dei progetti.
Parte delle informazioni relative alla ricerca si trovano su The Prime Pages, il sito sui numeri primi gestito dal Prof. Chris Caldwell dell'Università del Tennesse at Martin.


1. Calcolo di primalità


2 * 205833 * (22976221 - 1 ) + 1
Possible prime factor of the double Mersenne number MM36 (222976221-1-1).
Unfortunately, this prime number of 895,938 digits does not divide MM36.
20th August 2017 (895,938 digits).

2 * 764648 * (2859433 - 1 ) + 1
Possible prime factor of the double Mersenne number MM33 (22859433-1-1).
Unfortunately, this prime number of 258,722 digits does not divide MM33.
26th July 2017 (258,722 digits).

2 * 1972572 * (2756839 - 1 ) + 1
Possible prime factor of the double Mersenne number MM32 (22756839-1-1).
Unfortunately, this prime number of 227,838 digits does not divide MM32.
28th June 2017 (227,838 digits).

2 * 1630328 * (2756839 - 1 ) + 1
Possible prime factor of the double Mersenne number MM32 (22756839-1-1).
Unfortunately, this prime number of 227,838 digits does not divide MM32.
18th May 2017 (227,838 digits).

2 * 1491308 * (21257787 - 1 ) + 1
Possible prime factor of the double Mersenne number MM34 (221257787-1-1).
Unfortunately, this prime number of 378,638 digits does not divide MM34.
20th June 2014 (378,638 digits).

2 * 544952 * (21257787 - 1 ) + 1
Possible prime factor of the double Mersenne number MM34 (221257787-1-1).
Unfortunately, this prime number of 378,638 digits does not divide MM34.
20th June 2014 (378,638 digits).

2 * 1799828 * (2216091 - 1 ) + 1
Possible prime factor of the double Mersenne number MM31 (22216091-1-1).
Unfortunately, this prime number of 65,057 digits does not divide MM31.
10th June 2014 (65,057 digits).

2 * 1879484 * (2216091 - 1 ) + 1
Possible prime factor of the double Mersenne number MM31 (22216091-1-1).
Unfortunately, this prime number of 65,057 digits does not divide MM31.
11th June 2014 (65,057 digits).

2 * 9488 * (22976221 - 1 ) + 1
Possible prime factor of the double Mersenne number MM36 (222976221-1-1).
Unfortunately, this prime number of 895,937 digits does not divide MM36.
10th March 2014 (895,937 digits).

2 * 140340 * (21257787 - 1 ) + 1
Possible prime factor of the double Mersenne number MM34 (221257787-1-1).
Unfortunately, this prime number of 378,638 digits does not divide MM34.
10th January 2014 (378,638 digits).

226795 + 57083
Prime digit with no special form, certified and verified with Primo.
22nd November 2009 (8,067 digits).

61876115 * [RSA200]512 - 1
where RSA200 is the 200 - digit RSA factorization challenge number.
4th June 2005 (102,128 digits).

32476090*C(1531785651*210110+18,23)-1
Distributed search for a provable prime of the form p = k*C(x,23)-1 where C is the binomial function. Factoring p+1 to 30.9% and sieving by Jens Kruse Andersen. Proving p+1 factors with Primo by Pierre Cami, Luigi Morelli and V. M. Ulyanov. PRP tests with PrimeForm by the 4 mentioned and Décio Luiz Gazzoni Filho. PRP found by Pierre Cami. Konyagin-Pomerance primality proof by David Broadhurst.
June 2005.

25987968300*[RSA-200]512-1 and 49334180280*[RSA-200]512-1
A distributed search for a prime of the form k*[RSA-200]^512-1 which can only be proven prime by using the factorization of RSA-200, the product of 2 random 100-digit primes. Writing and running sieve to 10^12 by Jens Kruse Andersen. PRP tests by Jens Kruse Andersen, Göran Axelsson, Pierre Cami, Phil Carmody, Chuck Lasher, Predrag Minovic, Luigi Morelli, Martin Speirs, V. M. Ulyanov, Paul Underwood and Cedric Vonck. Finding and proving primes by Luigi Morelli and Paul Underwood.
May 2005.

22055611200*(10999+203959)#/(10999)#-1
A distributed search for a prime on the form p = k*(10999+203959)#/(10999)#-1, where the 101 smallest titanic primes divide p+1. Proving p+1 factors, writing and running sieve to 10^12 by Jens Kruse Andersen. PRP tests by David Broadhurst, Pierre Cami, Phil Carmody, Ken Davis, Predrag Minovic, Luigi Morelli and Paul Underwood. Finding and proving prime by David Broadhurst.
January 2005.

708477982733*3*5*11*13*19*29*37*53*59*2231290+1 (69649 cifre) 27 gennaio 2004 15:53:00 CDT - Payam number

3ˇ 2727699-1 (219060 cifre) 23 gennaio 2004 15:59:21 CDT - 321Search

210885 . 2295335-1 (88911 cifre) 25 novembre 2003 15:36:49 CDT - 15K search

1650292215 ˇ 2140588-1 (42331 cifre) 19 agosto 2003 18:04:12 CDT - 15K search

1650292215 ˇ 2127041-1 (38253 cifre) 19 agosto 2003 18:04:12 CDT - 15K search

1650292215 ˇ 2123255-1 (37113 cifre) 19 agosto 2003 18:04:12 CDT - 15K search

 


2. Fattorizzazione


(38ˇ10174-11)/9 = 401 ˇ 1531 ˇ 2333 ˇ 11103172231868232665208082632227[32] ˇ C134
C134 = P41 ˇ P93
P41 = 38399983418304274738274024976546740584967[41]
P93 = 727788294601551888099993904173706931765025315072705833126180244134209183779214775385936665507[93]

(Luigi Morelli / GGNFS 0.77.1 + msieve 1.39 for P41 x P93 / Mar 17, 2009 / 134 digits)

(4ˇ10174+41)/9 = 33 ˇ 7629737812981[13] ˇ 31668315658185358241[20] ˇ 64583320974668012969282589628685014327[38] ˇ C104
C104 = P44 ˇ P60
P44 = 13572177173944141379230865642040566250314827[44]
P60 = 738367971619864478723854016085408967268073560203846001125847[60]

(Luigi Morelli / GGNFS 0.77.1 + msieve 1.39 for P44 x P60 / Jan 4, 2009 / 104 digits)

(34ˇ10174-43)/9 = 23279 ˇ 344848243 ˇ 80308935953[11] ˇ 685748829763[12] ˇ 41268157890120643[17] ˇ C123
C123 = P53 . P70
P53 = 20911043547722046862836516832845716851816899295541089[53]
P70 = 9902032167234716303717452177033910330401475786201296097689113927382553[70]

(Luigi Morelli / msieve 1.38 for P53 x P70 / 23.84 hours / Dec 1, 2008 / 123 digits)

c 14535 = 31 . 101 . 599 . 9721 . 21860034500477033057[20] . 5024561734078743614019475672210465536770797634435009651390623763133271404343125421517983291123932203[c100] c100 = 95615984081251482246538695683308609617345441049 . 52549391007773634411218693704643056941771279438002147
(Luigi Morelli / msieve 1.30 for P47 x P53 / QS / Dec 18, 2004 / 100 digits)

2607-3 = 55 . 19 . 263 . 1401457092703061[16] . 254605005861515627419[21] . 1107375018476949027223897414997873720048902633664994028034804453[77]
p77 # Luigi Morelli and Alexander Kruppa SNFS 16 Dec 2004 - 98 digits.

c 14536 = 34 . 20411 . 9900799 . 194677459885211[15] . 8996108210139462978073236775295867026748488312730764496163714592360959433517967920644604084256992347[c100] c100 = 866509691540844271532732996709594633227[39] . 6071749876355753270908403958358496758132678092506191524465563[61]
(Luigi Morelli / msieve 1.30 for P39 x P61 / QS / Dec 13, 2004 / 100 digits)

 


3. Ricerca fattori GFN e xGF (Fermat numbers)


    New factors  k.2^n + 1  of  a^(2^m) + b^(2^m)                 

			  a   b      m           k          n      Year       Digits
			------------------------------------------------------------
 			  3   1    46352        197909    46353    2011       13,959
			  5   1   109737          5921   109739    2010       33,039
			  5   3    48370        129669    48371    2011       14,567
			  5   3    90011         23559    90013    2013       27,101
			  5   3    98971         19159    98974    2013       29,799
			  5   4    93834         12339    93837    2013       28,252
			  7   2    14392      28371113    14393    2017        4,341
			  7   2    46064        163211    46067    2011       13,873
			  7   6     6413       4828195     6414    2010        1,938
			  7   6   105719          4491   105720    2010       31,829
			  8   1     6659       4362423     6661    2010        2,012
			  8   1    49501        140739    49503    2011       14,908
			  8   3   109604          2261   109605    2010       32,998
			  9   5    14097      20085603    14098    2017        4,252
			  9   5    14498      21704897    14499    2017        4,372
			  9   5    93119         25447    93120    2013       28,036
			 10   1    71634         13633    71636    2010       21,569
			 10   3    92482         18357    92483    2013       27,845
			 10   7   105723          3963   105725    2010       31,830
			 11   1     6720       9596875     6722    2010        2,031
			 11   4   107259          2493   107261    2010       32,293
			 11   6     6658       5922527     6659    2010        2,012
			 11   7    99621         18551    99627    2013       29,995
			 11   9    46959        152641    46960    2011       14,142
			 12  11    92561         15783    92565    2013       27,870
			 12  11   105308          4631   105309    2010       31,705
			 12  11   107583          6699   107590    2010       32,392
			


4. Ricerca prime gaps

Con prime gap si intende la differenza tra un numero primo ed il successivo.
L'ennesimo prime gap, definito gn o g(pn) è la differenza tra il numero primo (n + 1)-esimo e l' n-esimo.

Il risultato seguente è stato ottenuto attraverso il team PGS (Prime Gap Sequences), organizzato su MersenneForum.
Il team comprende (in stretto ordine alfabetico):
S. Cole, L. Desnogues, R. Gerbicz, D. Jacobsen, A.P. Key, L. Morelli, A. Nair, C.E.L. Pinho, M. Raab, T. Ritschel, R.W. Smith (coordinator), D. Stevens.
I programmi per la ricerca al computer, in C and Perl, sono stati sviluppati da Robert Gerbicz, Dana Jacobsen, Antonio P. Key ed ottimizzati da tutti gli utenti.

La ricerca è iniziata ad aprile 2017 ed è tuttora in corso.
Maggiori informazioni si possono trovare sul sito del Dr. Thomas Nicely.


                    
				=========================================================================================
				 Gap  Cls   Discvrer  Year  Merit Digts  Following the prime
				=========================================================================================
				1450  CFC   LMorelli  2017  33.16    19  9808299410025809701
			


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