PODCAST OVERVIEW

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!

Keep in touch with me on social media: 

Instagram: @mike.514
Twitter: @mikeyperes
Facebook: @themichaelperes
Linkedin: Michael Peres


PREFACE

Curt Landon Noll

The first complete sentence Landon Curt Noll spoke (age 2) was “How far is the Sun?” Landon remains an Astronomer to this day where his study is focused on the inner solar system: the zone in which we ride our wonderful planet Earth, and that is bookended by the Sun and Jupiter.

As Cisco’s Resident astronomer Landon Curt Noll focuses on Balanced Technical Computing (BTC) by day, and focuses on our inner solar system as an Astronomer by night.  Landon has made astronomical observations during total solar eclipses from every continent and from every ocean on Earth.  He served as the expedition scientist for a team that searched for meteorites in the Antarctic ice and near the South Pole for a number of years.

Landon Curt Noll is the ‘N’ in the widely used FNV hash. He is also the founder of the International Obfuscated C Code Contest. He was a member of the working group that developed the IEEE POSIX standard.  He participated in the IEEE 1619 development of XTS and co-authored the XTS-AES Cryptologica paper on the security of Ciphertext Stealing. He serves as the Chair of the Co-operative Computing Award advisory panel to the Electronic Frontier Foundation, advising them on awards for the discovery of astronomically large prime numbers.

As a mathematician, he developed or co-developed several high-speed computational methods and as held or co-held eight world records related to the discovery of large prime numbers.  He is credited in Wikipedia as the co-inventor (with John Horton Conway) of a system for naming numbers of any size.  Landon graduated from Linfield College with a BA in Math/Physics. He is a member of the American Mathematical Society and is an associate of the American Astronomical Society.

For additional information about Landon see:

Prime Number 

A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That’s six factors. If we list the factors of 29, we only have 1 and 29. That’s two factors. So we say that 29 is a prime number, but 28 isn’t.

Another way of saying this is that a prime number is a positive integer that is not the product of two smaller positive integers.

Note that the definition of a prime number doesn’t allow 1 to be a prime number: 1 only has one factor, namely 1. Prime numbers have exactly two factors, not “at most two” or anything like that. When a number has more than two factors it is called a composite number.

Here are the first few prime numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.

Euclid’s Proof of Infinitely Many Prime Numbers 

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. 

Palindromic Primes 

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

235711101131151, 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.

Mersenne Primes

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 3731127, 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.

Millenium Problems 

Experiment and computer simulations suggest the existence of a “mass gap” in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.

The prime number theorem determines the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper, it asserts that all the ‘non-obvious’ zeros of the zeta function are complex numbers with real part 1/2.

If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution.

This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.

The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown.

In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston’s geometrization conjecture. Perelman’s proof tells us that every three manifold is built from a set of standard pieces, each with one of eight well-understood geometries.

Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles’ proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three.

The Lucas–Lehmer test

The Lucas–Lehmer test works as follows. Let Mp = 2p − 1 be the Mersenne number to test with p an odd prime. The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than Mp.

In pseudocode, the test might be written as

// Determine if Mp = 2p − 1 is prime for p > 2
Lucas–Lehmer(p)
    var s = 4
    var M = 2p − 1
    repeat p − 2 times:
        s = ((s × s) − 2) mod M
    if s == 0 return PRIME else return COMPOSITE

Performing the mod M at each iteration ensures that all intermediate results are at most p bits (otherwise the number of bits would double each iteration). The same strategy is used in modular exponentiation.

The Lucas test

In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known.[1][2] It is the basis of the Pratt certificate that gives a concise verification that n is prime.

 

 


ABOUT MICHAEL PERES
Michael Peres is a serial tech entrepreneur, software engineer, mathematician, and radio host. Michael is currently the CEO and founder of four successful tech start-ups, two media organizations, and a technical science podcast, the Michael Peres Podcast. Michael also provides software engineering expertise to a portfolio of hundreds of successful companies, start-ups, and prominent figures. Michael carries multiple degrees, including Computer Science, Mathematics, and Data Communications, and he is currently pursuing a M.S. in Biomedical Engineering.
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