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
The Teacher of Righteousness
in the Qumran Texts
By F. F. BRUCE, 1957
I. THE TEACHER AND THE TEXTS
‘The Teacher of Righteousness’ is the name given in a number of the lately discovered Qumran documents to a man who was held in high veneration by the religious community on whose beliefs and practices these documents have thrown so much light. If he was not actually the founder of the community, it was certainly he who impressed upon it those features which distinguished it from other pious groups which flourished among the Jews during the last two or three centuries of the Second Commonwealth. So far as we can gather from our present sources of information, he is never referred to by his personal name in the Qumran texts.1
The title bestowed on him by his followers, ‘The Teacher of Righteousness’ (Heb. moreh sedeq or moreh hassedeq), may echo Hosea x. 12, where the prophet calls to his people: ‘break up your fallow ground: for it is time to seek the LORD, till he come and rain righteousness (Heb. yoreh sedeq) upon you.’ The RV margin gives ‘teach you righteousness’ as an alternative translation to ‘rain righteousness Continue reading
MATE MASIE -“what I hear, I keep” wisdom, knowledge, prudence
Ntesi-Matemasi symbolizes wisdom and knowledge. “Nyasa bun mu nne mate masie masie.”
ME WARE WO -“I shall marry you” commitment, perseverance Continue reading
Tradition has it that Adinkra, a famous king of Gyaman (now part of Cote d’Ivoire) angered the Asantehene, Bonsu-Panyin, by trying to copy the Golden Stool. Adinkra was defeated and slain in an ensuing war. It has been suggested that the art of adinkra came from Gyaman. It is also significant that adinkra means farewell, or good-bye, hence the use of the special cloth on funeral occasions (eyie), to say good-bye to the departed.
Adinkra aduru (adinkra medicine) is the stuff used in the stamping process. It is prepared by boiling the bark of badie together with iron slag. Originally the printing was done on a cotton piece lying on the ground. Today, raised platforms with sack coverings act as the printing table. The designs, cut on pieces of calabash with pieces of wood attached for handling, are dipped into the adinkera aduru, then stamped onto the cloth. Adinkra cloth is not meant Continue reading
Adinkra Cloth from afroetic.com
African Symbols: Adinkra
The Adinkra symbols were originally designed by “Asante” Craftsment of Ghana, West Africa.
The symbols embody non-verbal communicative and aesthetic values, as well as the way of life of the people who designed them.
The symbols are usually printed on cotton fabric to produce “Adinkra cloths,” which may be worn on such celebrative occasions as child naming, community durbars and funerary rituals.
Each of the symbols has its Asante name and an accompanying literal English translation.
ADINKRAHENE – “chief of adinkra symbols” greatness, charisma, leadership
This symbol is said to have played an inspiring role in the designing of other symbols. it signifies the importance of playing a leadership role.