"The abc conjecture refers to numerical expressions of the type a + b = c.
The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities a, b and c.
Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers—those that cannot be further factored out into smaller whole numbers:
for example, 15 = 3 × 5 or 84 = 2 × 2 × 3 × 7.
In principle, the prime factors of a and b have no connection to those of their sum, c.
But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c."
OK, thank you for taking the time to try and explain that to me, I think I sort of get it.
The fact is, exceptions to the rule of "c < d" are hard to find, so while one can appreciate that the theorem is hard to PROVE, as a matter of experiment, one would suppose that the exceptions "peter out" pretty quickly.
Wikipedia gives the example of 2^7 = 3 + 5^3, with 128 > 30, as one of these exceptions. Of course, these are very small numbers. The next example in this pattern that I found was 2^14 = 759 + 5^6, with 16384 > 7590 ... hmmm. Well, and then 2^28 = 24294831 + 5^13, with 268435456 > 242948310 .
It takes a little work, because bc doesn't do factors, although the bash shell will take you pretty high. So we've got real power at our fingertips these days.