7-9...........13-15........19-21..... 23-25,

A few days before I set out to write this, scrolling through my twitter feed I found, @AlgebraFact · " Chen’s theorem: There are infinitely many primes p such that p+2 is either prime or the product of two primes." This read a little differently than how I remembered it so I set about milking the net to refresh myself. I concluded that Chen's Theorem is a lot like the parallel postulate, it is written lots of different ways.

Here's another, Theorem(Chen): For any even integer h∈2Z, there exist infinitely many primes p such that p+h is either a prime or a semiprime. Ok, make h=2 and it's the same thing.

But then Wikipedia has " Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes)." Now these are really very different statements, at least to young students. It is a link that number theorists recognize in some of math's long unproven conjectures.

So let's go back a bit, to the early twentieth century, and a German Mathematician named Edmund Landau.
"At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about
prime numbers. These problems were characterized in his speech as "unattackable at the present state of
mathematics" and are now known as Landau's problems.

They are as follows:

Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?

Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?

Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?

Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n^2 + 1?

As of June 2022, all four problems are unresolved." *Wikipedia.

Wow, so number 2 says almost the same as the twitter post, with a slight opening for "nearly prime", one of the terms sometimes used for semi-primes, or composite numbers that are the product of two primes. (As a teacher I hope that the students reading this would know the the two primes need not be distinct, so 9 and 25 are still semi-primes.)

Number one sounds more like the Wikipedia definition of the term. Obviously these two conjectures are interrelated.
A little before Chinese mathematician Chen Jingrun, first wrote about this idea in 1966 , and expanded on his proof in 1973, in 1947, Alfréd Rényi had showed there exists a finite K such that any even number can be written as the sum of a prime number and the product of at most K primes. So Chen had reduced the number of factors of the non-prime from some unspecified K, to 2, and showed that and included that there are an infinite number of "nearly prime pairs" with any even separation. So there are infinitely many primes p such that p + 4 is prime, or a semi-prime; and there are infinitely many primes p such that p + 6 is prime, or a semi-prime; and p+8, p+10,.....

Number Theory folks have names for some of these pairings, Cousin primes differ by four, (7 and 11 for example), and sexy primes differ by six, and if one of two sexy primes has a twin that falls between the sexy pair, they are called a prime triplet, 7,

__11, 13__for example, or__17, 1__9, 23. Chen's theorem says that there must be infinite examples of cousin and sexy pairs with a prime followed by a prime or a semi-prime. Then in 2015, Tomohiro Yamada took away the "every sufficiently large even another number " of Chen's theorem and gave a definite limit. Every even number greater than \(e^{e^{36}}\) , which big. But in time someone will find a way to knock that "sufficiently large number" down a little, maybe a lot. But that is with "nearly twin primes", it seems like Goldbach's conjecture and the twin prime conjecture still rest heavily in the "unattackable at the present state of mathematics" stage.

But then, in my youth most folks said we could never prove Fermat's Last Theorem, and then we did... well, not me, but it was the same type of whittling away at it through the ages until Andrew Wiles, inspired by a book he found in the library at age ten, completed a thirty year search for the "impossible" proof. Maybe one of your students will learn about this pair of "impossibles", and surprise us all.

Why not expose them to Goldbach's conjecture; what even numbers are the sum of two primes? It seems like all of them, at least every one we try. It has been tested up to 4*10^18 though, and so far, so good. But there are still great things to explore, how many ways can even n be written as the sum of two primes, and by what rules. 10 = 5+5 = 3+7 = 7+3 are the last two different? What if we don"t allow doubles like 5+5? If we want each sums primes to be distinct, 24 is the smallest even number expressible as the sum of two primes in three ways... and no, I'm not telling you, find them.

Just as I was writing this, one of the Fields Medalist winners in 2022 was presented to Oxford Professor James Maynard for his work on primes. One of his recent works about distributions of primes was toward a proof that there are an infinite number of prime numbers that do not have a 7 among their digits. He also cited the twin prime conjecture as one of his favorites.

Now what's the smallest number that is expressible in four ways, five...

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