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.
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
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
Temple of Maat – Tehuti
by (Anpu Ausar)
The paradigm for the works of R. A. Schwaller deLubicz has become known as the “symbolist perspective.” It is one of the keys to understanding the knowledge of Kemet. It should be considered a way (not the only way) to unlock our minds from the western, linear, left-brain, modern, evolutionary paradigmatic matrix that has us trapped. So many of us are struggling to ‘get it’ yet find ourselves crashing through one door only to be confronted by another door. Read the following: Volume 2, Chapter 23 of The Temple of Man.
Temple of Man – Volume 2
by R.A. Schwaller de Lubicz
Chapter 23 The Architectonics of the Pharaonic Temple
The master builder said to the disciple:
“You come from the earth, it has nourished you, and you will return to the earth. This element holds and keeps other elements. Continue reading
The soroban (算盤, counting tray) is an abacus developed in Japan. It is derived from the Chinese suanpan, imported to Japan around 1600. Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electronic calculators.
The soroban is composed of an odd number of columns or rods, each having beads: one bead having a value of five, called go-dama (五玉, “five-bead”) and four beads each having a value of one, called ichi-dama (一玉, “one-bead”). Each set of beads of each rod is divided by a bar known as a reckoning bar. The number and size of beads in each rod make a standard-sized 13-rod soroban much less bulky than a standard-sized suanpan of similar expressive power. Continue reading
A 13-month Lunar Calendar with Solar Dates
by Karl Palmen (2006)
I have found a relatively simple way of [adding] a solar date to a date of a 13-month lunar calendar and then found a simple way using this solar date system to regulate the 13-month lunar calendar.
The Lunar Calendar in The Invisible Landscape suggests a lunar calendar where each lunar year has 13 lunar months. The mean year would be just under 384 days.
I found that with such a calendar it would be quite easy to add a solar date to each such lunar calendar date to indicate the time of year. The solar dates belong to solar years and solar months defined thus: Continue reading