Skip to main content

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.

 





Comments

Popular posts from this blog

Great Chain of Being

One of the points that Lovejoy makes in the book of that title I mentioned last week is the importance, in the Neo-Platonist conceptions and in the later development of the "chain of being" metaphor, of what he calls the principle of plenitude. This is the underlying notion that everything that can exist must exist, that creation would not be possible at all were it to leave gaps.

The value of this idea for a certain type of theodicy is clear enough.

This caused theological difficulties when these ideas were absorbed into Christianity.  I'll quote a bit of what Lovejoy has to say about those difficulties:

"For that conception, when taken over into Christianity, had to be accommodated to very different principles, drawn from other sources, which forbade its literal interpretation; to carry it through to what seemed to be its necessary implications was to be sure of falling into one theological pitfall or another."

The big pitfalls were: determinism on the on…

A Story About Coleridge

This is a quote from a memoir by Dorothy Wordsworth, reflecting on a trip she took with two famous poets, her brother, William Wordsworth, and their similarly gifted companion, Samuel Taylor Coleridge.



We sat upon a bench, placed for the sake of one of these views, whence we looked down upon the waterfall, and over the open country ... A lady and gentleman, more expeditious tourists than ourselves, came to the spot; they left us at the seat, and we found them again at another station above the Falls. Coleridge, who is always good-natured enough to enter into conversation with anybody whom he meets in his way, began to talk with the gentleman, who observed that it was a majestic waterfall. Coleridge was delighted with the accuracy of the epithet, particularly as he had been settling in his own mind the precise meaning of the words grand, majestic, sublime, etc., and had discussed the subject with William at some length the day before. “Yes, sir,” says Coleridge, “it is a majestic wate…