Prime numbers
numbers is GIMPS, which was founded in 1996 and relies on thousands of volunteers, who offer their personal computers in order to contribute computational power. Even with this collaboration, verification of a single prime can still take weeks of energy-consuming computation, and with these primes having limited use past cryptography, it raises the question of whether the substantial investment of energy, time and funding is proportionate to the benefits provided for encryption systems, such as RSA. Prime numbers also have other modern applications, a major one being in error detection and correction in coding theory, which preserves the accuracy of data in both storage devices and communication channels. Primes are used here, as data are treated as numbers and then divided by a chosen prime, which generates check values to be used to see whether any of the original information has been altered. CRCs, for example, employ polynomials whose length corresponds to a certain prime, which minimizes the chance of an error going undetected. Another example is in hashing algorithms, where primes are implemented to help distribute data in order to decrease the likelihood of two distinct inputs giving the same output. So, although none of these applications require record-breaking large primes, they do still demonstrate the value of prime numbers in modern technology. Whilst outlining the use of primes in modern technology, we must also look at their limitations and the diminishing practical returns associated with the search for ever-larger prime numbers. The main problem with these huge primes is their limited practical use, with the GIMPS project admitting in a statement in 2016 that the new prime they found of over 22 million digits was ‘too large to currently be of practical value’. RSA tends to operate with primes in the order of hundreds of thousands of digits, as opposed to millions. As a result of this, prime numbers that have been found through intense searches, such as the GIMPS project, have never been implemented into real-world security systems. Furthermore, the costs of the searches for these redundant primes are considerable, with computing projects consuming large amounts of electricity over weeks or months, with a significant initial investment into hardware still required. Therefore, the opportunity cost of devoting vast resources to finding larger primes is significant, as this effort could instead be redistributed into fields with more tangible benefits, such as medical simulations or climate models. You could also argue that the motivation behind the search for ever-larger prime numbers comes from the mathematical achievement associated with it, which leads to the focus shifting away from technological development and towards a symbolic contest. Considering this, it becomes harder to justify the large-scale use of time, funding and energy where these resources could arguably be reallocated. With all this said, there is a significant argument for the search for ever larger prime numbers for the advancement of pure mathematics and computational methods. History shows that curiosity-driven research has often led to breakthroughs. For example, number theory itself was originally considered theoretical but now holds value in modern cryptography. As a result of the large projects in search of a larger prime number, technology has been advanced with the development of highly efficient algorithms and computational methods, which can now be deployed in other scientific and engineering fields. There is also the element of inspiration which comes from these successful large prime searches, which can spur mathematicians to develop new related projects. So, although these large prime numbers may remain largely impractical, the search for them has pushed forward technology and the human understanding of numbers. For these reasons, the case can be made that although there may be
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