∀ n∈ℕ, n > 1 ∃ p∈ℙ: n < p < 2n

SCANALYZER Tracking with Closeups
, , ,
1

“Huh?”, you say. Well, this is how mathematicians (or at least those enamoured of compact and cryptic notation) say

For all positive integers n greater than one, there is always a prime number p greater than n and less than two times n.

or, properly typeset, without the limitations of DIscourse topic titles:

\forall n \in \mathbb{N}, n>1, \exists p \in \mathbb{P}:n<p<2n

This is called Bertrand’s postulate, originally stated as a conjecture in 1845 by French mathematician Joseph Bertrand, who verified its correctness for all integers between 2 and 3,000,000.

In 1852, the Russian mathematician Pafnuty Lvovich Chebyshev proved the conjecture for all n>1, in what is now called the Bertrand–Chebyshev theorem. Chebyshev’s proof in terms of binomial coefficients is complex and messy, involving manual inspection to prove the assertion for all n\leq 468, but it gets the job done.

In 1919, Srinivasa Ramanujan found a much simpler proof based upon the properties of the Gamma function in one of his first publications.

Over the years, further generalisations have been proved including, in 1973 that there is always a prime between 3n and 4n, in 2006 that there is always a prime between 2n and 3n, and in 2011 that as n increases toward infinity, the number of primes between 3n and 4n also tends to infinity.

For what is this useful? Nothing, but it’s cool.

7 Likes
2

I wish I were able to appreciate its coolness. This leads me to posit: is the ability to understand maths at this level congenital or acquired? When in college math, there were one or two guys in a class of 25 or so, to whom mathematical proofs seemed to come naturally, effortlessly.

To me, it just seemed impossible to learn this stuff. I drew a blank, every time. Even if I memorized a given proof, it never made sense (Although I believe some of my thinking skills have improved with time/experience, I am afraid to try learning a little of this stuff today, because other cognitive skills - especially abstractions - have definitely deteriorated).

So, are math skills genetic of environmental?

1 Like
3

Y’know, I was a whiz at geometry! I could see the proofs at a glance. I never missed! Whereas for the rest of math it was a misery to me from first grade when we had to add up a whole page of just, like, 3+ 5. Why?
I was and am grateful for algebra, which is like phonics math. Step by step I could do it.
As for this post?
All I can say is, wow! Cool! I’ll take your word for it.

1 Like
4

My guess is that it’s a bit of both. I suspect that mathematics at the highest levels of achievement is something like the very pinnacle of the legal profession. The super-lawyer needs to have absorbed, assimilated, and internally organised a vast amount of information on statutes, case law, and historical evolution of the law, all of which requires not just talent but a large and diligent investment of time reading and understanding all of that material, but when it comes to crafting a new theory or finding a novel argument on behalf of a client, there is a native talent in putting together seemingly disconnected pieces in a new way which probably can’t be taught.

The mathematician must know a vast amount of work done over centuries upon which to draw in formulating and proving theorems, and this is especially the case today where many narrow specialties which were thought to be isolated and disconnected have been found to have a deep unity and connections. But when it comes to thinking of a novel way to approach a problem that has stumped the very best minds for centuries, I also think that’s something that can’t be taught and a few rare people can just do, and probably can’t explain how.

I completely lack that talent, so I cannot imagine what it must be like to have it. I can follow genius proofs in mathematics and appreciate the leaps that allowed their construction, but cannot begin to comprehend how I might make them myself or how it feels when you see the pieces fit together.

2 Likes
5

What is so fascinating, and seductive, about many problems in number theory is that they can be stated so easily a child (at least one who understands what things like “prime number” mean, which you can explain in less than a minute to anybody who can do division) can understand them, which are clearly true or false, and yet have defied the most brilliant mathematicians to prove or disprove, in some cases for centuries.

One of my favourites is Goldbach’s conjecture.

Every positive even integer can be written as the sum of two primes.

Well, they don’t get much simpler than that, do they? This one dates from 1742, and neither Christian Goldbach nor Leonhard Euler, who were present at its formulation, nor any of the hundreds of mathematicians, among the best-known names in the field, nor the tens of thousands of amateurs have managed to put a dent in it in the two hundred and eighty years since. Some weaker results have been proved, such as a proof in 1995 that every even number n\geq 4 is the sum of at most six primes, but as far as scaling the peak to two, nobody has gotten close.

Computer tests have shown that the Goldbach’s conjecture is correct for all numbers n\leq 4\times 10^{18}, but that means nothing in terms of proof for the endless sea of integers.

3 Likes