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Number Theory/Distribution of Primes

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Question
I've found an answer to a problem I've been working on since college, though the answer unearths more questions. I am in need of an individual with knowledge on the subject of primes and an open mind. The sieve of Eratosthenes aptly works wonders with primes, though the method itself eliminates the patterns made. For example, if a person looks simply at the pattern they'll find that the pattern of the first four primes repeats after 210. However when we reach the fifth and take it into account, the pattern grows to 2310 and repeats. Put the fact that the pattern being so strongly fought to see is ever-growing, if one were to stop taking next prime into account, one see's the pattern. There are other predictable topics associated using this line of thinking, but I'm in need of a person fluent on the subject that I can discuss this with.  I have prepared a paper explaining in detail the entire subject. If one has a legitimate idea and detailed explanation, would you humor me with a couple minutes of your time?

Answer
Hello Joel
First of all, the first 4 primes are 2,3,5,7.  Since there is only one even prime, this pattern can never repeat.
The main theorem on distribution of primes is that of Dirichlet.  It says that every arithmetic progression of odd numbers contains an infinite number of primes.  So it is not surprising you can find some patterns that repeat.  But it will be impossible to calculate where the next occurrence of that pattern will be.
However, if you send me a copy of your paper, I'll study it when I have finished exam marking.
Regards

vijilant

Number Theory

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Vijilant

Expertise

Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

Experience

Teacher of math for 53 years

Organizations
AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

Publications
Journal of mathematics and its applications M500 magazine

Education/Credentials
BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

Awards and Honors
State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 50 prize First class honours in OU BA Mathematics

Past/Present Clients
I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.

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