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.
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
American Sign Language Alphabet
American Sign Language (usually abbreviated as ASL or Ameslan) is used mainly in the US, Canada, and some parts of Mexico by an estimated number of 200.000 to 2 million. The American Sign Language is a manual language expressed not with combinations of sounds but with combinations of hand movement and shape, arms and body, and facial expressions. It is used natively and predominantly by the Deaf and hard-of-hearing.
Below is a table of American Sign Language Alphabet, also soon there will be available hand expression sentences and phrases to be used with deaf People to be able to communicate with them better. 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
Numbers in mythology
In Kemet, certain numbers were considered sacred, holy, or magical.
Three: symbol of plurality
The basic symbol for plurality among the ancient Egyptians was the number three: even the way they wrote the word for “plurality” in hieroglyphics consisted of three vertical marks ( | | | ). Triads of deities were also used to signify a complete system. Examples include references to NTR Atum “when he was one and became three” when he and his wife gave birth. Shu and wife Tefnut then gave birth to Ausar and his wife Auset, and Set and Nebthet. Ausar, Auset and son Heru would form the last NTR triad.
- The beer used to trick Sekhmet soaked three hands into the ground.
- NTR Ra is named three times to define the sun: dawn, noon, and evening. Continue reading