Primes, Modular Arithmetic, and Public Key Cryptography
(April 15, 2004)
Every cipher we have worked with up to this point has been what is called a symmetric key cipher, in that the key with which you encipher a plaintext message is the same as the key with which you decipher a ciphertext message. As we have discussed from time to time, this leads to several problems. One of these is that, somehow, two people who want to use such a system must privately and secretly agree on a secret key. This is quite difficult if they are a long distance apart (it requires either a trusted courier or an expensive trip), and is wholly impractical if there is a whole network of people (for example, an army) who need to communicate. Even the sophisticated Enigma machine required secret keys. In fact, it was exactly the key distribution problem that led to the initial successful attacks on the Enigma machine.
However, in the late 1970’s, several people came up with a remarkable new way to solve the Continue reading
Prime Arithmetic Progression
An arithmetic progression of primes is a set of primes of the form for fixed and and consecutive , i.e., . For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.
It had long been conjectured that there exist arbitrarily long sequences of primes in arithmetic progression (Guy 1994). As early as 1770, Lagrange and Waring investigated how large the common difference of an arithmetic progression of primes must be. In 1923, Hardy and Littlewood (1923) made a very general conjecture known as the k-tuple conjecture about the distribution of prime constellations, which includes the hypothesis that there exist infinitely long prime arithmetic progressions as a special case. Important additional theoretical progress was subsequently made by van der Corput (1939), who proved than there are infinitely many triples of primes in arithmetic progression, and Heath-Brown (1981), who proved that there are infinitely many four-term progressions consisting of three primes and a number that is either a prime or semiprime. Continue reading
Top 20 Arithmetic Progression of Primes
by Chris Caldwell
The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection’s home page. This page is about one of those forms.
Definitions and Notes
Are there infinitely many primes in most arithmetic progressions? Certainly not if the common difference has a prime factor in common with one of the terms (for example: 6, 9, 12, 15, …). In 1837, Dirichlet proved that in all other cases the answer was yes:
- Dirichlet’s Theorem on Primes in Arithmetic Progressions
- If a and b are relatively prime positive integers, then the arithmetic progression a, a+b, a+2b, a+3b, … contains infinitely many primes.
Sacred Geometry in Ancient Egypt
By M. Gadalla
The Cosmic Geometry
Herodotus, the father of history and a native Greek, stated in 500 BCE:
Now, let me talk more of Egypt for it has a lot of admirable things and what one sees there is superior to any other country.
The Ancient Egyptian works, large or small, are admired by all, because they are proportionally harmonious and as such appeal to our inner as well as outer feelings. This harmonic design concept is popularly known as sacred geometry—where all figures could be drawn or created using a straight line (not even necessarily a ruler) and compass, i.e. without measurement (dependent on proportion only).
The principles of sacred geometry are of Ancient Egyptian origin, which constituted the basis of harmonic proportions, as evident in their temples, buildings, theology, …etc. The Ancient Egyptian design followed these principles in well-detailed canons. Plato himself attested to the longevity of the Egyptian harmonic canon of proportion (sacred geometry), when he stated, “the pictures and statues made ten thousand years ago…”
The key to divine harmonic proportion (sacred geometry) is the relationship between progression of Continue reading
16TH CENTURY MATHEMATICS
The cultural, intellectual and artistic movement of the Renaissance, which saw a resurgence of learning based on classical sources, began in Italy around the 14th Century, and gradually spread across most of Europe over the next two centuries. Science and art were still very much interconnected and intermingled at this time, as exemplified by the work of artist/scientists such as Leonardo da Vinci, and it is no surprise that, just as in art, revolutionary work in the fields of philosophy and science was soon taking place.
It is a tribute to the respect in which mathematics was held in Renaissance Europe that the famed German artist Albrecht Dürer included an order-4 magic square in his engraving “Melencolia I”. In fact, it is a so-called “supermagic square” with many more lines of addition symmetry than a regular 4 x 4 magic square (see image at right). The year of the work, 1514, is shown in the two bottom central squares. Continue reading
Dürer’s polyhedron: 5 theories that explain Melencolia’s crazy cube
By Günter M Ziegler, 2014
In 1514 the German artist Albrecht Dürer (1471-1528) created the copper engraving Melencolia I. It was immediately recognised as a masterpiece, not only because of its remarkably fine and detailed execution and unsurpassed shadings, but also because of its unusual symbolism. Dürer was proud of his creation, carefully produced prints on the best paper he could get and gave them away as a proof for his artistry. But he was clever enough not to give any explanations. And thus even now, after 500 years of study (and certainly more than 500 interpretations, books, research papers, artistic essays and even blog entries about the piece), the mystery remains, and makes the piece as fascinating as ever. Continue reading
77 Facts About the Number 7
No piece of paper can be folded in half more than seven times.
In the Bible, Joshua’s siege upon the walls of Jericho was conducted in silence save for the blasts of seven ram’s horns. On the seventh day of the assault, Joshua’s men made seven circuits around the walls and the walls fell following mass shouting at the end of the seventh horn-blast. Continue reading