The Big Picture In Modern Number Theory
Goldbach's Conjecture is one of the oldest unsolved problems in all of mathematics and has not been rigorously proved. My attention is focused on it because I'm currently reading, the novel "Uncle Petros and Goldbach's Conjecture" by Apostolos Doxiadis, which on its surface is about a man who devotes his entire adult life to proving it without success.
Goldbach's Conjecture, the unsolved Riemann's Hypothesis, the recently proved Fermat's Last Theorem, and a variety of similar proven and unsolved problems in number theory, collectively imply that there is far more structure to the properties of whole numbers (like their status as prime or non-prime numbers) than the process by which they are defined would necessarily imply, and that as a result, the realm of the possible results of mathematical problems that involve these numbers are considerably more narrow than one would naively believe them to be if their subtle properties were not known.
Number theory is currently in a state where there are a whole panoply of highly interrelated and highly constrained conclusions about the properties of numbers that appear to be true to an extremely great level of certainty based upon brute force numerical approximations and intermediate results towards proofs that have almost proved many of these theories but the efforts to prove these theories to date have holes.
Mathematicians have been unable to ground this body of mathematical theorems in a way that establishes that they are really correct, because nobody has had the half a dozen or so insights that would be necessary to make the conceptual leaps necessary to prove that these theorems definitively correct, and it is even theoretically possible that these theorems could be impossible to prove logically even if they are actually true in all cases.
These half a dozen or so missing insights are a Holy Grail that keeps number theorists going for long hours on obscure work year after year, because anyone who spends some seriously time studying these unsolved problems, looking at the overwhelming evidence that almost all of them must be true or very nearly true with a very narrow class of exceptions, and making a stab at trying to solve them, comes away with a deep conviction that the insights that we are missing as we try to piece together proofs could be profound insights of wide application on a par with notions like the germ theory of disease in medicine, or the unification of space and time in general relativity. The unsolved problems in the field are well enough defined and sufficiently interrelated that one could imagine a single modern day Leonhard Euler solving all of them in a few years of a single career, even if that genius mathematican ended up dying young as so many mathematical geniuses have historically.
In particular, one of the insights that is strongly hinted at, although it has not been proved, is that the prime numbers can be used as a basis set to very simply generate all other numbers through addition in much the same way that they can be used to generate all other numbers through multiplication, even though nothing used to define addition and multiplication and the set of all numbers makes this necessarily true in any obvious or trivial way.
Goldbach's Conjecture
One common way of stating Goldbach's Conjecture, actually stated by Leonhard Euler in 1742 in response to a letter from Christian Goldbach, which is maddening because this profoundly challenging unsolved problem of mathematics is so simply stated is that:
"Every even integer greater than two can be expressed as the sum of exactly two prime numbers."