Goldbach’s conjecture was first posed by Christian Goldbach to Leonhard Euler in a letter dated June 7, 1742.^{[1]}
Conjecture (Goldbach’s conjecture, 1742). (Goldbach)
Every even number greater than or equal to 4 is the sum of two primes,^{[2]} and every odd number greater than or equal to 7 is the sum of three primes.^{[1]} Also, every even number greater than or equal to 6 is the sum of two odd primes, and every odd number greater than or equal to 9 is the sum of three odd primes.
The clause regarding even numbers is sometimes called the binary Goldbach conjecture or strong Goldbach conjecture, while the clause regarding odd numbers is sometimes called the ternary Goldbach conjecture or weak Goldbach conjecture.
Goldbach considered 1 to be prime,^{[1]} as did most mathematicians in that day and age, but our modern exclusion of 1 from the primes^{[3]} and inclusion in the units instead, makes the conjecture neither easier nor harder to prove or disprove.
Goldbach representations
 See parts restricted to two primes for even numbers and three primes for odd numbers and parts restricted to two odd primes for even numbers and three odd primes for odd numbers.
Binary Goldbach representations
An odd prime and 2 add up to an odd number. Two odd primes add up to an even number.
Number of binary Goldbach representations
On its face, it makes sense for the conjecture to be true. Given an even number
, there are
ways to express it as a sum of two odd numbers (distinct or not), and there are
ways to express it as a sum of two distinct odd numbers. There are only two ways to express
8 as a sum of two distinct odd numbers, and one of those ways is a pair of odd primes
(3 + 5). There are
8192 ways to express
32768 as a sum of two distinct odd numbers; it seems rather improbable that after examining all
8192 ways we would fail to find at least one that was a pair of primes.
In fact, the number crunching that has been done so far suggests that as
gets larger there is a slowly rising lower bound for the number of different ways of choosing odd primes
and
such that
.
^{[4]} Henry Fliegel and Douglas Robertson have plotted the
number of Goldbach representations for
even numbers up to
10000: their graph shows that, for the range reviewed, as
gets larger, the minimum number of Goldbach representations for
also gets larger.
^{[5]}
Finding a
prime gap from
to
would disprove the conjecture: it almost goes without saying that
(where
and
may be equal); therefore, if we can’t match any prime
up to
to one prime
equal or above (that is, to find a value of
that is prime), the conjecture is refuted. But when
Pafnuty Chebyshev proved that there is always a prime between
and
(see
Bertrand’s postulate), that did not automatically prove the Goldbach conjecture: that proof does not rule out the possibility that there is an
such that every
, for every prime
, is either
1 or a
composite number. This mismatch is farther reduced by the observation of
Ramanujan primes which effectively states that there is more than one prime between
and
for
.
The exclusion of 1 from the primes for the purposes of this conjecture only affects those even numbers that are one more than a prime. Requiring the primes to be distinct affects singly even numbers that are twice a prime. But given a large singly even number with millions of potential Goldbach representations, discarding one or two possibilities makes very little difference. Considering only odd primes also makes very little difference.
Expected number of binary Goldbach representations
This section of the article is under construction.
Please do not rely on any information it contains.
The number of odd primes in the upper half
is asymptotically equal to the number of odd primes in the lower half
, i.e.


Since we consider even
, it means that
is congruent to
0,
2 or
4 modulo
6. Except for
2 and
3, all the primes are congruent to
1 or
5 modulo
6.
Asymptotically, for each of the
odd primes (essentially all congruent to
1 or
5 modulo
6) in the lower half, the probability of matching an odd prime in the odd numbers of the upper half is

if
is congruent to 0 modulo 6, and

if
is congruent to
2 or
4 modulo
6, assuming a random distribution for the odd
prime numbers, since, knowing that the first number is chosen to be prime (most likely congruent to
1 or
5 modulo
6),
 if is congruent to 0 modulo 6, it is the sum of a prime and an odd number congruent to 1 and 5 modulo 6 or 5 and 1 modulo 6;
 if is congruent to 2 modulo 6, it is the sum of a prime and an odd number congruent to 1 and 1 modulo 6; or 5 and 3 modulo 6 (however, the only prime of the form 3 modulo 6 is 3);
 if is congruent to 4 modulo 6, it is the sum of a prime and an odd number congruent to 5 and 5 modulo 6; or 1 and 3 modulo 6 (however, the only prime of the form 3 modulo 6 is 3);
and the probability of finding a prime among numbers congruent to 1 or 5 modulo 6 is 3 times higher than among all the congruences modulo 6.
Therefore the expected number of Goldbach representations is asymptotically

if
is congruent to
0 modulo
6, and

if
is congruent to
2 or
4 modulo
6, assuming a random distribution for the odd prime numbers.
Expected versus actual number of binary Goldbach representations
A002375 From Goldbach conjecture: number of decompositions of
into an
unordered sum of two odd primes.
Expected versus actual number of binary Goldbach representations

A002375



Expected

990 
164 
1980 
0


991 
59 
1982 
2


992 
64 
1984 
4


993 
116 
1986 
0


994 
60 
1988 
2


995 
84 
1990 
4


996 
118 
1992 
0


997 
53 
1994 
2


998 
56 
1996 
4


999 
112 
1998 
0


1000 
74 
2000 
2





A002375



Expected

9990 
906 
19980 
0


9991 
328 
19982 
2


9992 
326 
19984 
4


9993 
670 
19986 
0


9994 
360 
19988 
2


9995 
438 
19990 
4


9996 
868 
19992 
0


9997 
382 
19994 
2


9998 
334 
19996 
4


9999 
730 
19998 
0


10000 
462 
20000 
2



Why is the actual number of Goldbach representations 2 to 3 times the [calculated] expected number of Goldbach representations!?
not equal to
when
.
Where is the flaw in the above reasoning? Is this due to the fact that the prime numbers are not actually randomly distributed? The actual numbers are 2 to 3 times higher...!?
See “(however, the only prime of the form 3 modulo 6 is 3)” above.
==> WOULD SOMEONE PLEASE VERIFY THE ABOVE REASONING!
If we investigate Goldbach’s comet (see A002372/graph) we effectively observe that there are two similar parts to the comet, the values of the upper part being typically double the values of the lower part.
Scrutinizing the
bfile, we can see that the
number of Goldbach representations for even numbers
congruent to
0 modulo
6} have approximately twice as many representations as their neighbors (
congruent to
2 or
4, modulo
6). Also, if you look at
A002372/graph (
Goldbach’s comet), it seems to have two parts, the lower third of the comet being similar to its upper two thirds, for which the above reasoning would suggest that the lower third (related to partitions into two primes for even numbers congruent to
2 or
4 modulo
6) has twice the number of points of the upper two thirds (related to partitions into two primes for even numbers congruent to
0 modulo
6, with probabilistically twice the number of representations as for the lower third, hence the upper part is
of the whole comet, if the [lower and upper] parts don't overlap and are contiguous), which means that the lower third has four times the density of the upper two thirds (again, assuming no overlap).
Goldbach’s comet for the number of binary Goldbach representations
See
Goldbach’s comet in
A045917/graph (number of decompositions of
into unordered sums of two primes).
Ternary Goldbach representations
Number of ternary Goldbach representations
A068307 From Goldbach problem: number of decompositions of
into a sum of three primes (
).

{0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 3, 2, 5, 2, 5, 3, 5, 3, 7, 3, 7, 2, 6, 3, 9, 2, 8, 4, 9, 4, 10, 2, 11, 3, 10, 4, 12, 3, 13, 4, 12, 5, 15, 4, 16, 3, 14, 5, 17, 3, 16, 4, 16, 6, 19, 3, 21, 5, 20, ...}
A007963 Number of (unordered) ways of writing
as a sum of
3 odd primes (
).

{0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 6, 8, 7, 9, 10, 10, 10, 11, 12, 12, 14, 16, 14, 16, 16, 16, 18, 20, 20, 20, 21, 21, 21, 27, 24, 25, 28, 27, 28, 33, 29, 32, 35, 34, 30, 37, 36, 34, 42, 38, 36, 46, ...}
Expected number of ternary Goldbach representations
(...)
Goldbach's comet for the number of ternary Goldbach representations
(...)
Proof of the ternary Goldbach conjecture
H. A. Helfgott claims to have proved the ternary Goldbach conjecture. In a 2012 paper, he expounds on “minor arcs” for the Goldbach conjecture. This he followed up with a paper almost exactly a year later on “major arcs,” which builds on the previous paper. Helfgott’s proof has not yet been verified by others.
See also
Notes
 ↑ ^{1.0} ^{1.1} ^{1.2} Clawson (1996): p. 236.
 ↑ Thomas Koshy, Elementary Number Theory with Applications. Harcourt Academic Press (2002): p. 116.
 ↑ It is now considered as the empty product of primes, thus 1 has no prime factors.
 ↑ Clawson (1996): p. 238.
 ↑ Clawson (1996): p. 242.
References
 Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, New York and London: Plenum Press (1996).
 H. A. Helfgott, Minor arcs for Goldbach’s theorem, arXiv:1205.5252 [math.NT] (Submitted on 23 May 2012)
 H. A. Helfgott, Major arcs for Goldbach’s theorem, arXiv:1305.2897 [math.NT] (Submitted on 13 May 2013)
External links