Monday, April 25, 2016

Patterns in Primes

I've been enjoying Dr. Terry Tao's enlightening patter regarding prime numbers and their statistical distribution along the number line.  Important results stretch back thousands of year's, starting with a proof credited to Euclid (as so many are) that no greatest prime number exists.

The Fundamental Theorem of Arithmetic is a next one:  all natural numbers decompose into prime factors except 1, with primes being those with precisely one factor (almost-primes have two).

Terry breaks it down for us into the multiplicative and additive branches of study.  Multiplied primes have received more attention, historically speaking.  In the additive world, we're looking for arithmetic series, and the distribution of intervals.

Do we ever run out of twinned primes, primes only two apart, like 3 and 5, 41 and 43?

As of when these Youtubes were made, we know if we keep twins, cousins, and sexies together in the set, we'll never run out (that set is infinite), but there's still no recognized proof that just twins will occur infinitely often ("i.o.") though Terry suspects that they do.

Primes do get more sparse lets remember i.e. they do tend to spread out.

What if only sexies remain, once we're out far enough i.e. some lower bound exists after which twins will no longer occur?  This result is not much expected, but at this point is hard to rule out.

Many interesting results have been obtained, including that the number of primes between n and 2n approaches n/log(n) as n increases.  That's log to the base e.

This is called the Prime Number Theorem or PNT and was known by the 1800s.

As a programming challenge, why not explore this assertion empirically?  I will make that suggestion on mathfuture, where I'm data warehousing some new curriculum ideas.

I've been developing a so-called "lambda calculus" track for 9-12 grade level topics, shades of Hermann (sp?) on sci.math, way back in the 1990s (he was a huge lambda calculus booster, by which he meant something more hard core and formal in meaning, and to which I am not opposed).

Dr. Tao knows how to continue the history of number theoretic research into primes right up to the last minute, owing to his front row seat as an active contributor to this literature.