I am Michael Peres and welcome to my podcast.
During an interview with Google, writing an algorithm to identify prime numbers is a possibility. Prime number applications go way beyond an interview, like...
- Discovering an evolutionary secret behind Cicada's breeding patterns
- One of mathematics most elegant proofs
- Brilliant methods devised to identify prime numbers,
- The incredibly mysterious relationship between prime numbers and the Riemann hypothesis
- Understanding Diffie-hellman cryptography
- Quantum Applications for detecting prime numbers, and breaking modern day cryptography
- Intelligent methods for sending information across the cosmos
... and much more.
In this 3 part series, I sit down with Landon Noll, 8 time world record holder for the longest prime number.
Discover why prime numbers matter and how it has and will continue to revolutionize our world!
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Curt Landon Noll
Euclid’s Proof Euclid first proved that there exists an infinite number of primes. (300 BC) Assume there are finitely many primes. Let S be the set of all n primes, p 1 p 2 p 3 …p n Let q = p 1 p 2 p 3 …p n + 1. There must exist a prime factorization for q. We claim that this factorization does not contain an element of S. This would imply that q uses a prime outside the set S, thus there are in fact more than n primes. Assume that p i, 1 <= i <= n, is in the prime factorization of q. Then it must divide p 1 p 2 p 3 …p n + 1. Specifically p i | 1. But since primes are greater than 1, this is our contradiction.
A palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns. The first few decimalpalindromic primes are
- 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … (sequence A002385 in the OEIS)
Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11.
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century.
The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, … (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, … (sequence A000668 in the OEIS).
If n is a composite number then so is 2n − 1. (2ab − 1 is divisible by both 2a − 1 and 2b − 1.) This definition is therefore equivalent to a definition as a prime number of the form Mp = 2p − 1 for some prime p.