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.
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."
Note that the prime numbers in question may be identical, that the number one does not count as a prime number for this purpose, and that there may be more than one pair of prime numbers that meet this condition for a given even integer.
A set of integers whose sum is equal to a number is called a "partition" of that number, so Goldbach's Conjecture and some related ideas explored here all involve mathematical theorem about various kinds of partitions of various kinds of numbers.
Fermat's Last Theorem (that there are no sets of three integers, a, b, c and n greater than zero, for which a^n+b^n=c^n for any value of n other than two), which was proposed in 1637 by Pierre de Fermat, and proved by Andrew Wiles (with the assistance of Richard Taylor and many mathematical predecessors who proved steps intermediate to the final result) in 1995, is also a mathematical theorem about partitions.
So is a similar theorem, which was proved by Lagrange in 1770 that "any natural number can be represented as the sum of four integer squares" (a set that includes zero).
The conjecture has a number of corollaries (i.e. theorems that are implied if the conjecture is true, although not necessarily visa versa).
For example, Goldbach's Conjecture implies that:
"Every odd integer greater than five can be expressed as the sum of exactly three prime numbers." The prime numbers in question may be identical, the number one does not count as a prime number for this purpose, and there may be more than one triple of prime numbers that meet this condition for a given odd integer. This theorem is actually the theorem originally formulated by Christian Goldbach sometimes called the weak Goldbach conjecture and it directly implies that every even number n ≥ 4 is the sum of at most four primes (likewise, a determination that every even number is the sum of at most four primes is sufficient to establish this conjecture).
"Every odd integer greater than seven can be expressed as the sum of exactly three odd prime numbers." (i.e. it is not necessary to all three prime partitions that include the number two).
"Every even integer greater than ten can be expressed as the sum of exactly six prime numbers."
"Every even integer greater than eighteen can be expressed as the sum of exactly six odd prime numbers."
"Every odd number greater than twenty-one can be expressed as the sum of exactly seven odd prime numbers."
A similar hypothesis, which is another conjecture, is that "Every large odd number (n > 5) is the sum of a prime and the double of a prime."
Progress Towards A Proof
A single even number greater than two that could not be expressed as a sum of exactly two primes, or a single odd number greater than five that could not be expressed as a sum of exactly three primes, would falsify these conjectures. But, no such number has been found in replicated computer searches up to 10^17.
Chen Jingrun proved "in 1973 . . . that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes)—e.g., 100 = 23 + 7·11."
Olivier Ramaré, in 1995 proved that every even number n ≥ 4 can be expressed the sum of at most six primes.
The Riemann Hypothesis
Another great unsolved problem in mathematics (i.e. a hypothesis that has neither been proved nor disproved), and in particular, in number theory, is the Riemann Hypothesis, and it too has interesting intermediate results and connections to the Goldbach Conjecture.
The Riemann Hypothesis is not obviously related to Goldbach's Conjecture, although it turns out there the two are related to each other. The Riemann Hypothesis concern the zeros of the Riemann zeta function.
The Riemann zeta function and a generalized version of it called the Dirichlet L-functions, are a based on special kinds of of infinite sequences of fractions that get generally smaller as the series progress. Since the fractions get smaller, the sum of these series often have finite value, and the Riemann zeta function and the various kinds of Dirichlet L-functions are equal to the sum of infinite series related to that particular function for a given complex number s.
The Riemann zeta function is the sum of an infinite series for terms 1/n^s where n in the series takes on each value from 1 to infinity, and s is permitted to be any complex number (i.e. a number with a real and imaginary component, either or both of which may be zero, where the imaginary component is equal to a real number times the square root of negative one sometimes called the "imaginary number"), rather than merely a natural number. Values of the zeta function for values of s equal to 3/2 (quantum theory), 2 (prime number frequency), 3 (physics and number theory) and 4 (physics) all have importance as physical or mathematical constants; the value of s equal to 1 is called the "harmonic series" with wide mathematical applications, and the value of s equal to 0 is -1/2.
The Riemann Hypothesis, which has neither been proven nor disproven, hypothesizes that all zero values of the Riemann zeta function are complex numbers of the form 1/2+it where i is the imaginary number and t is a real numbered variable multiplied by the imaginary number in the overall complex number put into the Riemann zeta function. Godfred Harold Hardy proved that there are infinitely many zeros of the Riemann zeta function on the line 1/2+it in the complex plane (although not all such values are zero) in 1914, but his proof didn't rule out the possibility that the Riemann zeta function might have other zeros.
The prime number theorem, imprecisely stated, provides that "the average gap between consecutive prime numbers near N is roughly ln(N)." The prime number theorem, in turn, is equivalent to a statement that there are no zeros of the Riemann zeta function of the form 1+it.
The generalized Riemann Hypothesis, conceived in 1859 by Bernhard Riemann with reference to work done by Johann Peter Gustav Lejeune Dirichlet in 1831, applies to generalized Riemann zeta functions called Dirichlet L-functions. Dirichlet L-functions which are sums of series of fractions with denominators identical to the Riemann zeta function, but for which the Riemann zeta function's numerator of "1" is only the simplest of cases. The generalized version also allows numerators that are functions of the "n" value for the term in question that can cycle through values such as 0, 1, -1, i, -i, w, w^2, and -w, -w^2 (where w equals e^(pi*i/3)), for successive terms of the series in various sequences defined in a particular way.
If the complex numbers s that are zeros of all Dirichlet L-functions, and not just Riemann zeta functions, are also all on the line 1/2+it in the complex plane, than the generalized Riemann hypothesis is true and a great many particularly strong conclusions in number theory can be reached, including conclusions related to the distributions of prime numbers, conclusions related to Goldbach's Conjecture, and conclusions related to how fast computing algorithms will work that are tighter than conclusions that can be made without proving this hypothesis.
Implications For Goldbach's Conjecture and Related Conjectures
The Riemann Hypothesis is important in number theory because Euler proved the product of the term 1/((1-p)^-s) for each prime number p and a given complex number s is equal to the Riemann zeta function for a given complex number s. Thus, the Riemann zeta function turns out to have a relationship of the proportion of numbers that are prime numbers, and that probability that a number of any given magnitude is a prime number.
The Riemann Hypothesis implies if true that every odd integer can be expressed a sum of not more than five primes, a result proven by Leszek Kaniecki in 1995, which limits violations of Goldbach's Conjecture to odd integers which can be expressed as a sum of one, two, four or five primes, but not three primes.
In 1997, Deshouillers, Effinger, te Riele and Zinoviev proved that the generalized Riemann hypothesis implies that "Every odd integer greater than seven can be expressed as the sum of exactly three odd prime numbers."