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Prime Numbers I



All right. Let's really put our geek hats on and talk about prime numbers.

A little over a year ago I wrote about an enjoyable evening I spent watching the musical Fermat's Last Tango, a fictionalized (and lyrical) presentation of Andrew Wiles' successful effort to prove that Fermat was right about a certain famously generalized form of the Pythagorean theorem.  For purposes of the musical, Wiles is renamed Daniel Keane.

Anyway, one of the conjectures that has acquired a good deal of importance in elite math-geek circles since Wiles' success is something called the "bounded gap conjecture" concerning prime numbers.

As a refresher, a prime number is any number higher than 1 that can be divided only by 1 and itself.

There are lots of prime numbers amongst the lowest counting numbers, but they thin out as one gets into the higher ones.

1 is by stipulation not a prime. Two and 3 are both primes. The first integer above 1 that isn't a prime, then, is 4. Five is a prime, 6 isn't. And so forth. No even number higher than 2 can be a prime of course, because it will necessarily be divisible by 2.

There are lots of "twin primes," that is, prime numbers that are separated from each other only by two.

For example, 5 and 7 are twin primes. So are 11 and 13.

There exist an infinite number of primes. But ... do there exist an infinite number of twin primes as well?

As one gets into the higher numbers, and primes themselves thin out, so do twin primes. But since 1849, due to the formulation of one Alphonse de Polignac, [that's him above], there has been a widespread conjecture to the effect that twin primes will never entirely die out either. There exist an infinity of such pairings.

Proving Polignac's conjecture would be a big deal in math.

And that still has not been done. BUT a piece of this puzzle has fallen into place.

Yitang Zhang has just established that there are infinitely many pairs of prime numbers that differ at most by 70 million. That may not sound very impressive at first gulp -- after all, 70 million sounds like a lot. A heck of a lot more than 2! So how does this "bounded gaps" theorem help us get to the Polignac conjecture?

I suppose that with all of infinity within which our conjecturing minds can roam about, and to someone with an abstract turn of mind, the difference between one finite number and another may seem a matter of detail. Seventy million is a finite number, and from here the direction of progress is straightforward, lowering the number X in the statement: "I have proven that there are infinitely many pairs of primes within at most X of one another."

A further comment, from another POV, tomorrow.

 





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