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LA DIMOSTRAZIONE DELL'IPOTESI DI RIEMANN È UNO DEI SETTE PROBLEMI DEL MILLENNIO

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LA DIMOSTRAZIONE DELL'IPOTESI DI RIEMANN È UNO DEI SETTE PROBLEMI DEL MILLENNIO

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Il CMI (Clay Mathematics Institute, con sede nel New Hampshire, negli Stati Uniti) ha messo in palio un milione di dollari per la soluzione di ciascuno di sette problemi.   L'Ipotesi di Riemann è uno di questi.

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The Clay Mathematics Institute | Clay Mathematics Institute

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A conference to celebrate the foundation of the Clay Mathematics Institute in 1998 and its contributions to the international mathematical community over the past 20 years. The conference will highlight the outstanding work of some of the mathematicians whose research CMI has supported.

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Dimostrare l'Ipotesi di Riemann forse permetterebbe di trovare dove sono i  numeri primi, con conseguenze devastanti!!!!

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LA CRITTOGRAFIA E' UTILIZZATA FIN DALL' ANTICHITA'

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LA CRITTOGRAFIA E' UTILIZZATA FIN DALL' ANTICHITA'

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La CRITTOGRAFIA è una tecnica che permette di proteggere le comunicazioni e le transazioni commerciali.

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La CRITTOGRAFIA è una tecnica che permette di proteggere le comunicazioni e le transazioni commerciali.

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COME SI USANO I NUMERI PRIMI PER LA CRITTOGRAFIA?

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COME SI USANO I NUMERI PRIMI PER LA CRITTOGRAFIA?

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L'utilizzo dei numeri primi in Crittografia avviene per via del sistema RSA (dal nome dei suoi tre inventori: Ronald Rivest, Adi Shamir e Leonard Adleman, che lo hanno ideato nel 1976).
Esso consiste nel prendere due numeri primi p e q molto grandi e nel moltiplicarli per ottenere n=p*q.
Il numero n è reso pubblico e viene usato per "codificare" i messaggi.
I numeri p e q sono invece privati e servono per "decodificare"  .
Il bello è che, se n ha almeno 300 cifre, per scoprire quali sono p e q, con le conoscenze attuali, servirebbero milioni di computer e centinaia di anni di tempo.

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L'enigma dei numeri primi (M. Du Satoy) - crittografia

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Uploaded by Nicola Chiriano on 2016-12-11.

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Lo scopritore del numero primo più grande conosciuto si è aggiudicato un premio di 3000 dollari.
Ma per i prossimi traguardi c’è una ricompensa molto più alta: per il primo numero primo con oltre 100 milioni di cifre l' Electronic Frontier Foundation di San Francisco ha messo in palio 150.000 dollari.

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I numeri primi di Mersenne conosciuti sono  solo 49. Gli ultimi 15 li ha trovati la GIMPS

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I numeri primi di Mersenne conosciuti sono  solo 49. Gli ultimi 15 li ha trovati la GIMPS

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Previous Day Stats

First Prime Tests

323

Verified Prime Tests

539

Newly Factored

103

All exponents below 57 727 781 have been tested and verified.
All exponents below 104 815 069 have been tested at least once.

First-time Lucas-Lehmer Testing Ends

One year ago, first-time PRP primality testing with proofs was introduced. It has been a huge success, saving GIMPS tens of thousands future double-checks. Going forward, the server will no longer make available exponents for first time Lucas-Lehmer tests. Users that have not yet upgraded to prime95 version 30.3 or gpuowl for GPUs should do so. Failure to upgrade will result in necessary double-check work. GIMPS has a multi-year backlog of double-checks to work through. There is even a chance that a new Mersenne prime is hidden in all those double-checks.

The server will continue to accept Lucas-Lehmer results. There is no need to worry about any LL tests that are currently underway.

BIG Changes Are Here! Prime95 version 30.3 released.

For almost 25 years, GIMPS has looked for new Mersenne primes by running a primality test on one computer and later running the exact same primality test on another computer to guard against hardware errors having corrupted the first primality test.

A breakthrough by Krzysztof Pietrzak makes it possible to eliminate the second primality test! The first primality test produces a proof file that can be securely verified with less than 0.5% of the work required to re-run the primality test. This breakthrough will nearly double GIMPS' throughput in the long run.

Version 30.3 is now available with PRP proofs. While not a required upgrade, at some point in the future only users running version 30.3 with PRP proofs will be assigned first-time primality tests. Should you run into any problems, support is available at this thread at Mersenne Forum.

Many thanks to Mihai Preda for discovering the paper on the breakthrough and realizing its importance to the GIMPS project. Also, thanks go to Pavel Atnashev for important ideas on adapting the discovery for use by GIMPS.

GIMPS Shirts For Sale!

Show your love of math and prime numbers with a GIMPS shirt! Friends and family will surely be impressed. Viliam Furik has taken the initiative to create, manufacture, and sell GIMPS polo and T-shirts. Viliam will not profit from the sale of the shirts. Any "loose change" from price rounding will be donated to GIMPS. Shirts are sold for the price of the shirt itself and embroidering. Custom text can be added upon request.

Price is expected to be about $9 for a T-shirt and $13 for a polo, plus shipping. Shipping ranges from $6.50 to $10 to Europe, USA, Argentina, Brasil, Canada, China, Egypt, Hong Kong, India, Israel, Japan, S. Korea, Mexico, Russia, Thailand, Turkey, and U.S. territories.

The pre-order/interest form is here.

The latest information on expected order date (we get a cheaper price if we order a batch of 50 or more), shirt sizing, etc. can be found here at MersenneForum. You can also ask questions of Viliam on the forum.

BOINC users with GPUs can now participate

SRBase has created a BOINC project to hand out trial factoring assignments on large Mersenne numbers. These are very quick work units. To get the most of the BOINC client without any issues, a FAQ at the SRBASE forum is available.

Note that there's no way to individually credit SRBase users in GIMPS trial factoring stats. All work is credited to GIMPS user SRBase.

GIMPS 2019 fundraiser (updated)

GIMPS is a victim of its own success! In 2008, after claiming the EFF award for discovery of the first 10 million digit prime, GIMPS had about $25,000 cash on hand to fund server hardware, ISP fees, and $3000 awards for discovery of new Mersenne primes. Due to the unexpected, but welcome, discovery of 6 new Mersenne primes since 2008, GIMPS now has enough cash to fund just two years of ISP fees (much less if a new Mersenne prime is discovered).

GIMPS needs your help! For the first time in our 23 year history we are conducting a fundraiser. Our goal is to raise $5000 or more in 2019 to build a better cushion for upcoming ISP expenses and cover the Please consider making a next prime discovery award. donation so GIMPS can continue its quest to discover ever larger Mersenne primes for years to come.

GIMPS has set up a new subforum where the board of directors will keep our users well informed of its decisions, past and present finances, and fundraising progress.

UPDATE: The 2019 fundraiser has been a huge success, easily exceeding our goal! Many thanks to all those that contributed. GIMPS now has a cushion to guard against unexpected occurrences such as multiple prime discoveries or a server hardware failure. Our ISP expenses of about $2000 per year continue.

Version 29.8 (build 6) is the latest version available for download.

Highlights of version 29.8 include

AVX-512 support.

Modified torture test dialog box with new options and better understanding of the L1/L2/L3 cache hierarchy.

More robust implementation of Gerbicz error checking in PRP tests. This replace LL testing as the default work type.

You can view the full list of changes in the version history file here.

51st Known Mersenne Prime Found!

December 21, 2018 - The 2 Great Internet Mersenne Prime Search (GIMPS) has discovered the largest known prime number, 82,589,933-1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018. The new prime number, also known as M82589933, is calculated by multiplying together 82,589,933 twos and then subtracting one. It is more than one and a half million digits larger than the previous record prime number.

GIMPS has been on amazing lucky streak finding triple the expected number of new Mersenne primes -- a dozen in the last fifteen years. This prime was even luckier for Patrick Laroche, striking pay dirt on just his fourth try. For years, Patrick had used GIMPS software as a free "stress test" for his computer builds. Less than four months ago he started prime hunting on his media server to give back to the project. By way of comparison, some GIMPS participants have searched for more than 20 years with tens of thousands of attempts but no success. This proves that, with luck, anyone can find the next new Mersenne prime.

The new prime is only the 51st known Mersenne prime ever discovered. Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers more than 350 years ago. GIMPS, founded in 1996, has discovered the last 17 Mersenne primes. Volunteers download a free program to search for these primes, with a cash award offered to anyone lucky enough to find a new prime. Prof. Chris Caldwell maintains an authoritative web site on the largest known primes, and has an excellent history of Mersenne primes.

Patrick is one of thousands of volunteers using free GIMPS software available at www.mersenne.org/download/. Credit for this prime goes not only to Patrick Laroche for running the Prime95 software, Woltman for writing the software, Blosser for keeping the Primenet server running smoothly, and the thousands of GIMPS volunteers that sifted through millions of non-prime candidates. In recognition of all the above people, official credit for this discovery goes to "P. Laroche, G. Woltman, A. Blosser, et al."

You can read a little more in the press release.

All tests smaller than M(43112609) have been verified, officially making it the 47th Mersenne Prime

April 8, 2018 - Nearly 9 years ago in August 2008, M(43112609) was discovered, and now GIMPS has finished verification testing on every smaller Mersenne number. With no smaller primes found, M(43112609) is officially the 47th Mersenne prime.

At the time of the discovery, M(43112609) was actually the 45th known Mersenne prime because M(37156667) wasn't discovered until 2 weeks later, and M(42643801) was found nearly a year later in June of 2009! The last time a Mersenne prime was discovered out of order was in 1988 when M(110503) was found over 4 years after M(132049), and in 1961 M(4423) was discovered mere seconds before M(4253) because of the order in which the printout was read.

This highlights the importance of waiting until all smaller exponents have been tested and verified before we can say definitively where any Mersenne prime is ranked. Due to the distributed nature of the project as a whole, numbers are not always tested in order and a smaller Mersenne prime may yet be found. For that reason, thanks to all the GIMPS members that contributed their resources towards achieving this milestone. Join now to help GIMPS press onward with verification tests to prove M(57885161) is the 48th Mersenne prime!

All tests smaller than M(42643801) have been verified, officially making it the 46th Mersenne Prime

February 22, 2018 - Nearly 9 years ago in June 2009, M(42643801) was discovered, and now GIMPS has finished verification testing on every smaller Mersenne number. With no smaller primes found, M(42643801) is officially the 46th Mersenne prime. Thanks to all the GIMPS members that contributed their resources towards achieving this milestone. Join now to help GIMPS press onward with verification tests to prove M(43112609) is the 47th Mersenne prime.

Verification tests are an important part of the GIMPS project. Errors can occur during a test of smaller numbers, invalidating the end result. Only by doing a double-check with matching results are we able to say for sure that a Mersenne number is composite. Until all numbers below a Mersenne prime have been verified, we don't know for sure if it's the 46th known Mersenne prime or if there might be a smaller one that we missed due to a machine error, so we at GIMPS celebrate these important verification milestones.

50th Known Mersenne Prime Found!

January 3, 2018 - Persistence pays off. Jonathan Pace, a GIMPS volunteer for over 14 years, discovered the 50th known Mersenne prime, 2 77,232,917-1 on December 26, 2017. The prime number is calculated by multiplying together 77,232,917 twos, and then subtracting one. It weighs in at 23,249,425 digits, becoming the largest prime number known to mankind. It bests the previous record prime, also discovered by GIMPS, by 910,807 digits.

Just how big is a 23,249,425 digit number? It's huge!! Big enough to fill an entire shelf of books totalling 9,000 pages! If every second you were to write five digits to an inch then 54 days later you'd have a number stretching over 73 miles (118 km) -- almost 3 miles (5 km) longer than the previous record prime.

Jonathan Pace is a 51 year old Electrical Engineer living in Germantown Tennessee. He is a long-time math enthusiast now working at FedEx and active in community charities. As SysAdmin for his charities, he runs Prime95 on all PCs and servers because GIMPS emails him if one doesn't check in, which is helpful for monitoring these remote computers from home or work. The PC that found the new prime took six days of intense computation on a quad-core Intel i5-6600 CPU to prove the number prime.

To be thorough, the prime number was independently verified with four different programs running on various hardware configurations.

In recognition of the individual discoverer, the software authors, the GIMPS project leaders, and every GIMPS participant's contribution, credit for the new prime goes to "Jonathan Pace, George Woltman, Scott Kurowski, Aaron Blosser, et al.".

Could you be the next lucky volunteer to discover a brand new Mersenne Prime? You'll need a reasonably modern PC and the free software on the download page.

You can read a little more in the press release.

49th Known Mersenne Prime Found!

January 7, 2016 - GIMPS celebrated its 20th anniversary with the discovery of the largest known prime number, 2 74,207,281-1. Curtis Cooper, one of many thousands of GIMPS volunteers, used one of his university's computers to make the find. The prime number, also known as M74207281, is calculated by multiplying together 74,207,281 twos then subtracting one. It has 22,338,618 digits -- almost 5 million digits longer than the previous record prime number.

Dr. Cooper is a professor at the University of Central Missouri. This is the fourth record prime for Dr. Cooper and his university. The primality proof took a month of computing on a PC with an Intel I7-4790 CPU. Dr. Cooper and the University of Central Missouri is the largest contributor of CPU time to the GIMPS project. The discovery is eligible for a$3,000 GIMPS research discovery award.

You can learn more in the press release or watch the standupmaths interview with Curtis Cooper regarding his discovery:

New Assignment and Recycling Rules

February 2014 - Since 2008, GIMPS has given users one year to complete assignments. This rule has not been enforced. This has held up completing milestones as some assignments did not complete even after several years.

During February 2014, new assignment and recycling policies were put in place to help GIMPS make steady progress on milestones by detecting assignments that are proceeding extremely slowly or not at all.

This affects users in two ways:

When they occasionally become available, if you want to test the smallest exponents you'll need to sign up on the assignment rules page and be aware of the shorter timeline for returning results.

Your computers that are proven producers will have 8 or 9 months to complete assignments. Your slower computers and computers with a limited track record will still have a full year to complete their assignments.

Users testing 100 million digit numbers are not affected by these new rules. Assignments made prior to March 1, 2014 will be given a year, as promised, to complete (plus a grace period if the assignment is close to complete).

More News and Discussions

GIMPS forums - Here you can chat with fellow GIMPS members, get help with installation questions, learn more about how GIMPS works, etc.

Make Math History!!

You could discover one of the most coveted finds in all of Mathematics - a new Mersenne prime number. We've found fifteen already. Join in on this fun, yet serious research project. All you need is a personal computer, patience, and a lot of luck.

In addition to the joy of making a mathematical discovery, you could win a (USD) $3,000 cash GIMPS Research Discovery Award for each Mersenne prime discovered, and the Electronic Frontier Foundation is offering a $150,000 award to the first person or group to discover a 100 million digit prime number! See how GIMPS will distribute this award if we are lucky enough to find the winning 100 million digit prime.

What are Mersenne primes and why do we search for them?

Prime numbers have long fascinated amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etc. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime of the form 2 P-1. The first Mersenne primes are 3, 7, 31, 127 (corresponding to P = 2, 3, 5, 7). There are only 51 known Mersenne primes.

GIMPS, the Great Internet Mersenne Prime Search, was formed in January 1996 to discover new world-record-size Mersenne primes. GIMPS harnesses the power of thousands of small computers like yours to search for these "needles in a haystack".

Most GIMPS members join the search for the thrill of possibly discovering a record-setting, rare, and historic new Mersenne prime. Of course, there are many other reasons.

Last Updated January 30, 2021

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marin mersenne prime numbers gimps primenet prime95

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Great Internet Mersenne Prime Search - PrimeNet

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Previous Day Stats First Prime Tests 323 Verified Prime Tests 539 Newly Factored 103 All exponents below 57 727 781 have been tested and verified.All exponents below 104 815 069 have been tested at least once. One year ago, first-time PRP primality testing with proofs was introduced. It has been a huge success, saving GIMPS tens of thousands future double-checks.

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CI SONO ANCHE ALTRI METODI... AD ESEMPIO I NUMERI DI MERSENNE (matematico francese del XVI secolo)

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CI SONO ANCHE ALTRI METODI... AD ESEMPIO I NUMERI DI MERSENNE (matematico francese del XVI secolo)

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I numeri primi di Mersenne sono numeri primi che si possono esprimere
nella forma: 2p-1 dove p è a sua volta un numero primo.
Non sempre, anche se p è primo, il numero 2p – 1 è primo, però a volte sì.

Per esempio, per p = 7, 27 – 1 = 127 è primo, mentre 2¹¹ – 1 = 2047 è

divisibile per 23 e per 89.

Oggi, per cercare i numeri primi si cercano principalmente quelli di questa categoria: si prende un numero primo p , si calcola il numero 2p – 1

e si verifica se è primo.

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Il metodo più antico è quello del crivello di Eratostene (III Sec.A.C.).
Se ad esempio si vogliono i numeri primi minori di 100, si scrivono i numeri da
1 a 100 e si eliminano prima i multipli di 2, poi i multipli di 3 e così via.
Alla fine rimarranno solo i numeri primi.

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Per semplicità è stato chiamato M74207281

Per leggerlo tutto occorrerebbero giorni, se scritto la sua lunghezza sarebbe di decine di chilometri. Fa parte dell'insieme dei numeri di Mersenne. Importante per la crittografia

La sua sigla è M74207281, ma in realtà viene usata solo per comodità, perché se si dovesse mai scrivere questo numero ci vorrebbero decine di chilometri e si impiegherebbero interi giorni per leggere le sue cifre dalla prima all'ultima. È il numero primo più grande che sia mai stato trovato, conta 22.338.618 cifre e ha anche un altro pregio, che i matematici sanno bene cosa significhi. Fa parte infatti dei numeri di Mersenne (dal nome del matematico che ne escogitò la formula), esprimibile cioè da 2 elevato a potenza (necessariamente un numero positivo e intero) meno uno. Nel caso specifico M74207281 (e da qui la ragione della sigla) è stato calcolato moltiplicando il numero 2 per se stesso per ben 74.207.281 volte e sottraendo 1 al risultato. Fino a oggi il progetto ha già consentito di identificare quindici numeri di Mersenne, su un totale di 49 conosciuti.

L'esperto di numeri primi

A trovarlo è stato ancora una volta il professor Curtis Cooper dell'Università del Missouri, che circa tre anni prima aveva trovato un numero primo e di Mersenne di 17 milioni di cifre. Cooper lavora all'interno di un progetto denominato Great Internet Mersenne Prime Search (Gimps), che utilizza i computer messi a disposizione dai volontari di tutto il mondo per elaborare calcoli facendo affidamento su un software disponibile per diversi sistemi operativi.

La crittografia

I numeri primi in generale, e quelli molto grandi in particolare, piacciono da sempre ai puristi della matematica, ma al di là dei sofismi esistono anche due ragioni più pratiche che motivano questo genere discoperte. Da una parte i ricercatori testano la potenza dei calcolatori, dall'altra il numero primo è da sempre corteggiato da chi si occupa di crittografia, in quanto custode dei sistemi di sicurezza più infallibili. Questi numeri vengono infatti usati nei sistemi di crittografia digitale asimmetrica, la più utilizzata in ambito informatico, che si basa sulla protezione dei dati attraverso il prodotto di due numeri primi molto grandi, che costituiscono le chiavi di decifrazione. Non a caso l'Electronic Frontier Foundation ha messo in palio 150 mila dollari per chi troverà un numero primo da cento milioni di cifre e 250 mila dollari a chi riuscirà a trovare un numero primo di un miliardo di cifre, proprio perché sarebbe una garanzia assoluta per la privacy.

Amore per la matematica

Chi non lo ha mai provato non può capire, eppure l'amore per la matematica a detta di chi ce l'ha è talmente puro da spingere i matematici a onorare i concetti più significativi di questa scienza. Da tempo infatti i matematici hanno pensato, per aumentare l'attenzione verso una materia considerata ingiustamente arida, di legare alcune date a una sequenza di numeriche possono far ricordare alcuni importanti concetti matematici. Così, dopo il Pi greco Day (14 marzo) che si celebra da alcuni anni, ecco aggiungersi il Giorno di Fibonacci (23 novembre). Forse ci vorrebbe una giornata dei numeri primi, che non potrebbe che cadere in un giorno che è un numero primo di un mese che è un numero primo.

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Scoperto il numero primo più grande, ha oltre 22 milioni di cifre

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Per semplicità è stato chiamato M74207281 La sua sigla è M74207281, ma in realtà viene usata solo per comodità, perché se si dovesse mai scrivere questo numero ci vorrebbero decine di chilometri e si impiegherebbero interi giorni per leggere le sue cifre dalla prima all'ultima.