Surely most readers no longer even remember that it is a prime number. The answer is simple, it is a number within the field of natural numbers (those that are integers and positive) greater than one, which is only divisible by itself and by one. For example, 2, 3, 5, and 7 are prime; 0, 1, 4, 6, 8 and 9 are not.
Even those who remember the subject probably never understood what they were for, beyond being something like mathematical entertainment. The reality, however, is that prime numbers intervene in our daily lives and increasingly, with countless practical applications in the field of security, computing and technology.
Specifically, they are fundamental in the development of security keys (forget the “password” of your bank account and think about the crypto world), search algorithms (Google, Brave, Mozilla, all depend on them) and data organization (the answers of Artificial Intelligence programs) and in countless theorems, such as Fermat’s (its resolution was key to cryptography and algebraic geometry) or Prime Numbers (used in cryptography, algorithm development and factorization in computing). , etc.)
Even with the advances in mathematics and science, there is much we do not know about prime numbers. For example, one of the doubts that has always existed is whether, being within the field of natural numbers, which is infinite, the prime numbers are finite or not (is their number limited?).
Although there are some formulas capable of generating some prime numbers (putting the cart before the horse), the best known is the Euler polynomial, (n^2+n+41) which goes up to the number 39, we do not have any capable to identify them all and/or to not generate “false cousins”.
The problem, and hence the advantages of its use, is that prime numbers do not follow any regular pattern, except that the distance between them would seem to widen as the numbers get larger, although the twin prime conjecture: “ There is an infinite number of primes p such that p + 2 is also prime”, violates this (a prize of $1,000,000 awaits anyone who solves the Riemann Hypothesis – Hilbert’s eighth problem).
Possibly Euclid was the first to define them in his “Elements” of the year 300 BC. Since then Cataldi, Descartes, Fermat, Leibniz, Euler, Turing, Landry, Lucas, Catalan, Sylvester, Cunningham, Pepin, Putnam, Lehmer and practically all the greats mathematicians in history studied them.
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In 1644 in his “Cogitata Physico-Mathematica” the monk Marin Mersenne developed a simple way of noting prime numbers, particularly the largest ones, by studying that natural numbers (n) generated prime numbers under the formula Mn=2^n – 1. These are the ones we know today as Mersenne Cousins. Note that this formula is just a simplified way of writing them down (we don’t even know if all primes can be defined this way).
Between 1876 Edouard Lucas discovered M127, a 39-digit number that would be the twelfth Mersenne prime number and the largest until 1951 (M61, M89 and M107 were discovered later, in 1883, 1911 and 1914).
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It must be understood that although there were some shortcuts (Euclid’s division by tests, Eratosthenes’s sieve or Fermat’s Test from 1640) that allowed us to identify if a number was possibly a prime number, until then the classical way of determining it It was to divide it by 1,2.3,…, n-3,n-2,n-1 and itself, a cyclopean job.
With the advent of computers and the development of new tests for large numbers (Miller-Rabin, the AKS algorithm, the ECPP, the Atkin sieve), we can say that the discovery of “primes” exploded: between 1952 and 1996 another 22 were identified (35 in total).
That year George Woltman created the GIMPS (Great Internet Mersenne Prime Search), a collaborative effort to try to identify the largest Mersenne prime in existence. Before the end of December, the Frenchman Joel Armengaud gave birth to M1,398,269, composed of 420,961 digits, used for the first time by a PC powered by a Pentium processor (previously supercomputers had been used) and since then every one or two years it has been discovering a new number, until 2018 when an impasse occurred, which had a lot to do with technology.
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Working independently (but based on the developments of Mihail Preda – the first to use GPUs in 2017 with his GpuOwl program -, G.Woltman and A.Blosser, among others), this October 12, Luke Durant, 36 years old , identified the Mersenne number 52, M136279841 with 41,024,320 digits (if printed, it would fill a 225-page book), winning the $3,000 prize that he donated to the mathematics department of the Alabama School of Mathematics and Science , awarded by the GIMPS.
It may not seem like much, but whoever discovers the first prime number with more than 100 million digits, possibly the next discovery (there are about 700 people actively looking for it, some for more than 10 years), something else awaits: a prize of u$d 150,000.
Please note that this number and its predecessors, M74207281, M77232917 and M82589933, continue to have a provisional ranking and there could be other “intermediate cousins” until they pass all the tests.
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The discovery of this former NVIDIA technician, which took almost a year, goes beyond a simple figure that ends the 28-year reign of Intel-based PCs in the handling and discovery of large prime numbers.
What it did was open the door to the use of GPUs (an NVIDIA A100 in Dublin, Ireland identified the number and it was confirmed by an H100 in San Antonio, Texas, using a Lucas-Lehmer test), which until now were basically used in crypto mining and the development of Artificial Intelligence, to the field of fundamental mathematics and scientific research creating a “supercomputer” in “the cloud”, using thousands of GPU servers distributed across 24 data center regions in 17 countries .
Ah!, for whoever discovers the first prime number of more than 1,000 million digits (it is not mandatory to discover the intermediate ones) the prize is $250,000, don’t miss it.
Source: Ambito
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