It’s very easy to say mathematician Christian Goldbach’s most famous statement, but it hasn’t been possible to prove it, nor has it been possible to disprove it. That’s why it’s a conjecture.
This conjecture states that every even whole number greater than 2 is the sum of two prime numbers. Symbolically, it’s represented as: p + q = 2n, where p and q are primes, and 2n, any even number greater than 2. It’s key to remember that a prime number is a number (other than 1) whose only divisors are 1 and itself, such as the number 3, which can only be divided by 1 and 3. By contrast, the number 4 isn’t a prime, because it doesn’t only have 1 and 4 as divisors, but also the number 2.
Neither humans nor computers
Goldbach was a mathematician who was born in 1690 in Königsberg, Prussia (now Germany), and died in 1764, in Moscow, Russia. His famous conjecture is still on the minds of mathematicians across the world, who are eager to know if it’s really true.
One of them is Galo David Ruiz Soto, a Philosophy of Science professor from the Faculty of Sciences at the National Autonomous University of Mexico (UNAM), who in an interview for Tec Review says that neither humans nor computers have been able to solve the conjecture.
“There have been certain approximations, but we don’t know as yet if the conjecture is true or false, or if it’s proven as a theorem, as we mathematicians say. This is partly because prime numbers are not fully understood; meaning there’s no general way of constructing them,” the academic explains.
Natural numbers are the numbers used to count, explains Ruiz Soto. They can be constructed by establishing number 1 as the base, and a successor, which means adding another 1. So, you start with the unit (1) and add 1 to it, then you get 2 (the successor of 1), to which you then add 1, and you get 3 (the successor of 2), and that’s how you build the rest, to infinity.
“Even numbers are easy to construct, because they’re of the form 2n. As long as a whole number can be multiplied by 2, the result will be an even number. Odd numbers are also very easy, because if you have 2n, you only have to add 1 to get the formula for all odd numbers (2n + 1). However, this doesn’t happen with prime numbers. There’s no general formula to say in advance what a prime number is going to be,” says Ruiz.
Many isn’t the same as every
The professor tells us that computer calculations have been made to find out if a particular even number is the sum of two primes, and the calculations have been made for many large numbers, but that’s not enough for mathematics.
“This would only be empirical evidence that the conjecture’s true for certain cases; however, general statements are what we’re interested in. The fact that we have thousands and thousands of empirical tests that certain even numbers are the result of the sum of two prime numbers isn’t enough to say that the statement is proven,” says the UNAM professor.
The most famous theorem (not conjecture) in mathematics is that of Pythagoras. This proposition says that the sum of the squares on the legs (cathetus) of a right triangle is equal to the square on the longest side (hypotenuse). Although it’s taught from middle school onwards, the proof of its universal validity (for all right triangles) should be seen in high school (this is not always the case in Mexico).
This professor insists that although Goldbach’s conjecture is apparently very simple, in reality, it’s complicated because it deals with prime numbers, whose behavior humanity hasn’t yet been able to precisely understand.
“They’re unpredictable. We have no way of knowing general things about them,” he says.
It so happens that everyday people believe that just a one-line explanation in colloquial language is needed in order to prove the validity of a conjecture like Goldbach’s, but this isn’t the case.
This specialist claims that it’ll most probably require an advanced mathematics paper approximately 200 pages long to prove Goldbach’s conjecture. That’s if it’s even possible, because so far, there have been no indications that it can be achieved.
To better illustrate this last idea, Ruiz recalls the viewpoint of another great mathematician born in Königsberg, David Hilbert, who in the 20th century, clarified the true mission of mathematics like no one else.
“He said that what matters isn’t the statement, but the demonstration. That’s where there’s genuine mathematical progress.”
Apostolos Doxiadis, a writer of Greek origin born in Australia, is a popularizer of mathematics. He became famous in 1992, when he wrote a novel about Goldbach’s conjecture.
It’s called Uncle Petros and Goldbach’s Conjecture, a book that tells the story of a young man in search of his true vocation and his uncle, a mathematical genius determined to show that every even integer greater than two is the sum of two prime numbers.
In this text, Doxiadis also recounts the lives of brilliant 20th century mathematicians, such as Kurt Gödel, Alan Turing, and Srinivasa Ramanujan.