The abc Conjecture: Motivation

Some of the earliest problems in mathematics asked about the integer solutions to simple polynomial equations. For instance, what are the possible right triangles with whole number side lengths? The solution dates at least back to the Ancient Greeks; the side lengths are related by Pythagoras' famous formula x² + y² = z². The 7th century Indian mathematician Brahmagupta studied integer solutions to the equation x² - 2y² = 1 as well as the same formula with 2 replaced by a general integer n (called Pell's equation). Many other similar equations have been studied for centuries or millennia.

In general, a Diophantine equation is a polynomial equation for which we are interested in integer solutions. Counterintuitively, some questions about solving these in the integers may be more difficult than considering all types of solutions. For example, the fundamental theorem of algebra states that any polynomial in a single variable has a root over the complex numbers (e.g. x³ - 4x² + 17x + 20 = 0 is true for some complex number). However, there is often no integer solution to such equations.

Historically, different types of Diophantine Equations were typically solved by ad hoc methods, as they come in many different varieties. However, one general observation that connects many of these equations is that they state something about the factorization of a sum of two numbers. Pythagoras' equation says something special about the sum of two squares, namely that it is another square! Similarly, Pell's equation says that one plus some number multiplied by a square has the property that it too is square. Our motivating question may then be taken to be:

How does the factorization of a sum of two numbers relate to the factorizations of the individual numbers?

The abc Conjecture provides a partial answer to this question. Its name comes from the fact that we are considering equations of the form a + b = c and asking how the factorizations of the three numbers relate. Mathematicians David Masser and Joseph Oesterlé first made the conjecture in 1985 while studying integer points on what are called elliptic curves, in this case given by the equation y² = x³ + k (where k is a fixed integer). This is yet another example of a sum having special factorization properties. Throughout the rest of this post, we will see how thinking about the motivating question might lead you to formulating the abc conjecture.

Simply put, we want the answer to our motivating question to be "it doesn't." Somehow, the additive and multiplicative structures of the integers should be independent of one another. This is in some ways a deep statement, and not at all intuitively clear, but we'll begin with this assumption. In other words, for an equation a + b = c, if all three numbers satisfy some special factorization properties (e.g. being cubes, etc.) it should in some sense be a coincidence. Our next task is to make this progressively less vague. First, we need a definition.

Definition: Two numbers are relatively prime if they share no common prime divisors.

For example, 34 and 45 are relatively prime, but 24 and 63 are not, because they are both divisible by 3. Here is how we will express our independence hypothesis: for any equation of the form a + b = c, where a and b are relatively prime, if a and b are divisible by high powers of primes, c almost always is not. This is in keeping with our theme because "divisible by high powers of primes" is special factorization property. That is, most prime factorizations should look more like 705 = 3*5*47 and not 768 = 2⁸*3. The assumption that a and b are relatively prime exists to rule out silly equations like

2ⁿ + 2ⁿ = 2^{n + 1},

in which all three numbers are divisible by arbitrarily high powers of 2. This doesn't represent some special connection between addition and multiplication - all we've done is multiplied the equation 1 + 1 = 2 by 2ⁿ. If we assert that a and b are relatively prime, then the prime factors of each of the three numbers are distinct, and we eliminate the uninteresting examples. Next, we require a mathematical notion that measures "divisibility by high prime powers".

Definition: The radical of a number n, denoted rad(n), is the product of the distinct prime powers of n. Also define rad(1) = 1.

For example, rad(705) = 3*5*47 = 705 (since the factors 3, 5, and 47 are distinct) but rad(768) = 2*3 = 6. The radical function forgets about any powers in the prime factorization, keeping only the primes themselves. Notice that the radical of a number can be as large as the number itself, but it can also be much smaller. The amount by which rad(n) is smaller than n can be taken as a measure of to what extent n is divisible by large prime powers.

Now we return to our equation a + b = c (where we will now consistently assume the relatively prime hypothesis). A reasonable way to test for high prime power divisibility for all three of these numbers is to calculate rad(abc) = rad(a)rad(b)rad(c) (the reader may wish to prove this last equation). Since rad(abc) could be as large as abc itself, it seems likely that rad(abc) would usually be much larger any of the individual numbers, the largest of which is c. For example, consider 13 + 22 = 35. In this case, rad(abc) = rad(13*22*35) = 13*2*11*7*5 = 10010, which is much larger than c = 35. However, this property does not always hold true. Consider another example, 1 + 8 = 9. Now we have rad(abc) = rad(1*8*9) = 2*3 = 6, and 6 < 9 = c. Notice that this anomaly reflects something weird going on; the equation can also be written 1 + 2³ = 3², so one plus a cube is a square. Testing different values of a, b, and c gives the impression that equations of the second sort are rare. Therefore, we make an almost mathematical conjecture:

"Almost" Conjecture: For equations of the form a + b = c where a and b are relatively prime, rad(abc) is almost always greater than c.

We're close! The equation rad(abc) > c is a bona fide mathematical condition that we can check. However, we have yet to render "almost always" into mathematical language. Clearly there are infinitely many a + b = c equations to look at. What does it mean to say that "most of them" behave in some way? We know from our 1 + 8 = 9 example that there are at least some exceptions. Maybe we could assert that there are less than 10 total exceptions, or less than 100. However, these numbers seem arbitrary, so we'll just guess that there are only finitely many exceptions. That is, all but at most N of these equations, for some fixed finite number N, satisfy our hypothesis. In conclusion, we conjecture that:

Conjecture 1: For all but finitely many equations of the form a + b = c where a and b are relatively prime, rad(abc) > c.

Finally, a real conjecture! Unfortunately, it's false. In other words, there are infinitely many such equations for which rad(abc) ≤ c. Don't worry! It's rare in mathematics to come up with the correct statement on the first try! In the next post, we'll prove our conjecture 1 false and see how to correct it.

Sources: https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The abc Conjecture, 12 sep 2013.