The ABC Conjecture probes deep into the darkness, reaching at the foundations of math itself. First proposed by mathematicians David Masser and Joseph Oesterle in the 1980s, it makes an observation about a fundamental relationship between addition and multiplication. Yet despite its deep implications, the ABC Conjecture is famous because, on the surface, it seems rather simple.
It starts with an easy equation: a + b = c.
a, b, and c, the variables that give the conjecture its name, have some restrictions. They need to be whole numbers, and a and b cannot share any common factors, that is, they cannot be divisible by the same prime number. So, for example, if a was 64, which equals 26, then b could not be any number that is a multiple of two. In this case, b could be 81, which is 34. Now a and b do not share any factors, and we get the equation 64 + 81 = 145.
It isn’t hard to come up with combinations of a and b that satisfy the conditions. You could come up with huge numbers, such as 3,072 + 390,625 = 393,697 (3,072 = 210 x 3 and 390,625 = 58, no overlapping factors there), or very small numbers, such as 3 + 125 = 128 (125 = 5 x 5 x5).
What the ABC conjecture then says is that the properties of a and b affect the properties of c. To understand the observation, it first helps to rewrite these equations a + b = c into versions made up of the prime factors:
Our first equation, 64 + 81 = 145, is equivalent to 26+ 34= 5 x 29.
Our second example, 3,072 + 390,625 = 393,697 is equivalent to 210 x 3 + 58 = 393,697 (which happens to be prime!)
Our last example, 3 + 125 = 128, is equivalent to 3 + 53= 27
The first two equations are not like the third, because in the first two equations, you have lots of prime factors on the left hand side of the equation, and very few on the right hand side. The third example is the opposite — there are more primes on the right hand side (seven) of the equation than on the left (only four). As it turns out, in all the possible combinations of a, b, and c, situation three is pretty rare. The ABC Conjecture essentially says that when there are lots of prime factors on the left hand of the equation then, usually, there will be not very many on the right side of the equation.
Of course, “lots of,” “not very many,” and “usually” are very vague words, and in a formal version of the ABC Conjecture, all these terms are spelled out in more precise math-speak. But even in this watered-down version, one can begin to appreciate the conjecture’s implications. The equation is based on addition, but the conjecture’s observation is more about multiplication.
“If it’s true, then it will be the most powerful thing we have,” says Peter Sarnak, professor at Princeton University.
It would be so powerful, in fact, that it would automatically unlock many legendary math puzzles. One of these would be Fermat’s last theorem, an infamous math problem that was proposed in 1637, and solved only recently by Andrew Wiles in 1993.Wiles did not solve Fermat’s Last Theorem via the ABC conjecture — he took a different route — but if the ABC conjecture were to be true, then the proof for Fermat’s Last Theorem would be an easy consequence.
(This short article has been published with prior permission of the above mentioned author)