How to win a million dollars from Faber & Faber?
Area of Math: Number Theory.
In 2000, the famous publishing house, Faber and Faber, publishers of novel ‘Uncle Petros and Goldbach’s conjecture’, offered a prize of 1 million dollars.
For what?
Nothing dramatic. In fact, for something quite simple.
We only need to prove that every even integer x > 2 can be written as the sum of two primes.
Also called, the Goldbach’s Conjecture.
That’s all.
This simple problem has remained unsolved since 1742 when Mathematician Christian Goldbach sent this conjecture in a letter to Leonard Euler.
(Well, he actually asked if every integer x > 2 can be written as the sum of three primes. But that was when he believed that 1 was also a prime number which is now an abandoned convention.)
So, it has been about 280 years since countless mathematicians have been trying to prove this conjecture. I know this conjecture is true, seeing as it has been manually tested up to 400,000,000,000,000. Although I, of course, haven’t been able to prove this. Getting those million dollars seems a wee bit harder now.
It isn’t impossible, though. Many people have proved conjectures around the Goldbach’s conjecture.
In 2013, Harold Helfgott proved that every odd integer n ≥ 5, is the sum of three prime numbers. This is known as the Weak or the Ternary Goldbach’s Conjecture. (The one I am discussing in this article is called the Strong Goldbach’s Conjecture. If the strong conjecture is proved to be true, then the weak conjecture will also be true.) The weak Goldbach’s Conjecture took 271 years to be solved. Hopefully, the strong one won’t take another 200 years to be solved.
Famous number theorists Hardy and Littlewood also conjectured that every odd number n ≥ 5, is the sum of a prime and twice a prime number. Nobody has been able to prove this conjecture either. Anyway, that’s besides the point.
Whether someone may or may not solve the Goldbach’s Conjecture, we can definitely learn about this problem and it’s background. So, let’s dive into this conjecture and maybe one of you out there would be able to solve one of the oldest, famously unsolved problem in Mathematics.
Christian Goldbach was born in 1690 in Königsberg. On the 7th June, 1742, he proposed this conjecture in a letter that he sent to Euler, another equally great mathematician.
After a few revisions, the conjecture now asks: “Is every even integer x > 2, the sum of two prime numbers?”
First, why even?
Well, that’s simple to answer seeing as all the prime numbers are odd except 2. (This conjecture does not include 2 anyway.) And we know that if we add any two odd numbers, the answer is always even. [(2n+1) + (2m+1) = 2(n + m + 1); n ≥ 0, m ≥ 0.]
We can now create a picture where we can visually show each and every prime number being added to each other.
We will do this by writing the prime numbers on either side of a triangle. Then, we will draw parallel diagonals from every prime number.
In all the intersections, we will write the sum of the prime numbers whose diagonals are intersecting. I have drawn the diagram till prime number 37 below:
**Since the conjecture is for integers x > 2, I have omitted the sums for the ‘2’ diagonal. But I have still added it in the picture above because it is after all a prime number and I didn’t want to discriminate against it.**
The above diagram can go on growing infinitely.
Now what are the things we can notice from the picture?
- First, it’s symmetric. We get the same numbers both sides of the centre. This is just because addition is commutative.
- Then, from top to bottom, every even integer x, such that, 4 ≤ x ≤ 34, are present. This shows us that every even integer 4 ≤ x ≤ 34, can be written as the sum of two primes.
- Next, most of the even integers are repeated more than once. That would mean there is more than one way to write those even integers as the sum of primes.
- Something that may not be initially clear, but would be as we keep extending and growing the above picture infinitely is that as the integers get bigger, more are the ways in which we can write them as the sum of two primes.
In fact, there is a way to find out the approximate number of ways we can write these integers as the sum of two primes: Using the Prime Number Theorem.
Prime Number Theorem
One of most important theorems in Number Theory, the PNT is about how many primes are there, less than a number n. So, for example, it answers the question how many primes are there less than, say, 83,729,345.
The formula says that the number of primes less than n is approximately n divided by natural log of n.
where π(n) is the prime-counting function defined as the number of primes less than n.
Assuming you are aware about the natural log function, we know if I plot log of x, it is actually growing, but it grows really, really slowly.
By taking the log of a big number, we are actually getting a very small number. For example, the natural log of 1 billion is only about 21. But, this small number is also getting bigger. Slowly, but surely.
In fact, it’s tending to infinity.
Note: The ‘~’ symbol used in the equation above is the Tilde symbol. In simple terms it just means that π(n) is similar to or approximately equal to n/ln(n). Technically, it means that if we divide π(n) with n/ln(n), it tends to 1 as n gets bigger or as n tends to infinity,
So, this tells us how many, on average — it gives us at least an approximation— of how many primes we can expect less than any number.
Now, we can get some results from this.
If say we have a box of 4 red marbles and 7 blue marbles and we want to find the proportion of red marbles in box, what do we do? We divide the number of red marbles by the total number of marbles. So, the proportion or probability would be 4/11.
In the same way, to find the proportion of primes less than a number n, we divide the number of primes less than n by n. Or,
Thus, the proportion of primes less than n is 1/ln(n).
We previously asked how many primes are there less than 83,729,345. We can now find the proportion of primes less than 83,729,345. The natural log of 83,729,345 is 18.2431000714 ≈ 18.243. So, the proportion of primes less than 83,729,345 is 1/18.243 ≈ 0.05481554568. Rounding this up, the proportion is about 5.48% or about 5.5%. Thus, we can say that about 5.5% of numbers less than 83,729,345 are prime.
This proportion is like finding the probability. That is, if I pick any random number less than 83,729,345, the probability of that number being prime is about 5.5%.
The larger the n, the smaller is the proportion.
What does this imply? It implies that the number of prime numbers are getting more rarer and spaced out, as the numbers get bigger.
Returning to the original topic of discussion
Using the PNT, we can estimate the number of ways to write a given number n as the sum of two primes.
We will use 2m instead of n (because according to the conjecture, we should take an even integer). How many ways can we write 2m, as p + q where p and q are prime?
Now, if we write 2m as the sum p + q, then one of p and q must be bigger than or equal to m and the other one then must be less than or equal to m. That is, we can take p ≥ m, q ≤ m.
Using the proportion concept in PNT explained previously in this article, if we look at a particular number, p, that’s greater than m and less than 2m, the probability of p being prime is
So the chance of p being prime is also 1/ln(m).
And that’s the same for the chance of q being prime. That is, if we look at q that’s lesser than m, the probability of q being prime is
So the chance of q being prime is 1/ln(m).
So we wrote 2m as p + q, where p ≥ m and q ≤ m. The chance p being prime is 1/ln(m) and the chance of q being prime is also 1/ln(m). So, the chance of both of them being prime at the same time is
The proportion of primes would be 1/(ln(m))² if they were independent events, but it’s not quite independent.
We now have to check how many chances we get. To compute this probability we had to choose an p between m and 2m. So there are m choices. Number of ways to write 2m as p + q is about equal to m times 1/(ln(m))²
If we take m as 1 million, then, the natural log of a million is about 14. So, the number of primes less than a million that can be written as p + q is 10⁶/14² = 10⁶/196.
m is a whole lot bigger than the ln(m). Or, the numerator is much, much larger than the denominator. We know that the natural log function grows very slowly. So the number of ways to write 2m as p + q = m/(ln(m))² is an enormously large number as m gets bigger much faster than ln(m).
Thus, for any given large number there are lots of ways of writing it as a sum of two primes.
Many have tabulated the things we have discussed in the form of a plot. In the graph, we plot each integer m versus the number ways it can be written as the sum of two primes. The number just keeps growing, as expected, and we get this amazing picture:
This is called the Goldbach’s Comet.
The Goldbach’s Comet in the above image is plotted till 10⁶. The plot is another way of confirming that the conjecture must be true as there is not one point in the graph that seems unusually off or out of place. The graph grows quite steadily.
After all this discussion, we believe that an even integer x > 2 could be written as the sum of two primes. We know that these integers can be written as the sum of two primes in multiple ways. And we also know that the larger the integer, the more are the ways in which it can be written as the sum of primes.
Well…there is no real conclusion to this article. Honestly, it’s quite unsatisfying. We are just making guesses. Good ones, but guesses nonetheless.
So, let’s see who the Goldbach’s conjecture will help turn into a millionaire.
