- #1
sachinism
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Find all prime numbers p that can be written p = x4 + 4y4 , where x, y are positive
integers.
integers.
CRGreathouse said:Factor x4 + 4y4.
robert Ihnot said:Factor, what, over the gaussian integers? In that case we know that a prime must be congruent to 1 mod 4. Also, other than 5, which is a solution, exactly one of the terms is divisible by 5, for the sum to be prime.
Petek said:x4 + 4y4 factors over Z.
Petek said:x4 + 4y4 factors over Z.
Prime numbers of given form are numbers that are only divisible by 1 and themselves. They are often expressed in the form of 2n+1, where n is a positive integer. These numbers have special properties and are used in various mathematical applications.
There are various methods for finding prime numbers of given form. One method is to use a sieve algorithm, which involves eliminating all non-prime numbers in a given range. Another method is to use modular arithmetic and check for patterns in the remainders when dividing by certain numbers.
Prime numbers of given form have many applications in mathematics and computer science. They are used in cryptography, coding theory, and number theory. They also have practical applications in creating efficient algorithms and data structures.
Yes, there are many famous prime numbers of given form. One example is the Mersenne primes, which are prime numbers of the form 2n-1. The largest known prime number, discovered in 2018, is also a Mersenne prime. Another famous example is the Fermat primes, which are prime numbers of the form 2^(2^n) + 1.
Yes, prime numbers of given form have practical applications in various fields such as computer science, cryptography, and data security. They are also used in creating efficient algorithms and data structures for solving real-world problems. For example, the RSA encryption algorithm uses prime numbers of given form in its encryption and decryption processes.