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This is: The Goldbach conjecture is probably correct; so was Fermat's last theorem, published by Stuart Armstrong on the AI Alignment Forum.
EDIT: Added a section on Euler's conjecture.
The Goldbach conjecture is likely
The Goldbach conjecture is that "every even integer above two is the sum of two primes". For example,
4
2
2
6
3
3
8
5
3
, and so on.
Though this is a mathematically precise statement, we can talk about the "probability" of it begin correct. How so?
Well, by the prime number theorem, the probability of a random number less than
N
being prime, is
1
log
N
. So if we sum up all the primes less than
N
, we get
N
log
N
2
different sums; these sums will be less than
2
N
So, is
N
itself is one of these sums? Well, the "probability" that it's not the total of any given sum is
1
−
1
2
N
; therefore the probability of it being the total of none of the sums is:
1
−
1
2
N
N
log
N
2
1
−
1
2
N
2
N
N
2
log
N
2
≈
1
e
N
2
log
N
2
So the probability of
N
being the total of such a sum is roughly:
1
−
e
−
N
2
log
N
2
Therefore, the probability of all numbers
N
being the total of such a sum is roughly:
p
2
∞
∏
N
2
1
−
e
−
N
2
log
N
2
Now, the infinite product
p
2
converges to a non-zero number if and only if the sum
∑
∞
N
1
e
−
N
2
log
N
2
converges to a finite number. That series can be seen to be convergent (for example, by noting that
e
−
N
2
log
N
2
1
N
2
for large enough
N
and using the comparison test).
If use computers to get an estimate of
p
2
, we get a pretty low probability. However, most of that improbability mass is on the low numbers, and the Goldbach conjecture has been tested up to
4
×
10
18
. So, if we assume it's valid up to
1000
, we numerically get:
p
1000
∞
∏
N
1000
1
−
e
−
N
2
log
N
2
≈
0.9961.
So the Goldbach conjecture is pretty likely, and, the more examples we discover where it holds, the more likely it is to hold all the way to infinity.
"Probabilities" of logical facts
The above reasoning seems dubious. The primes are not defined by random sampling among the natural numbers; quite to the contrary, they come from a mathematical rule of extreme precision. So what do these probabilities mean?
Let
X
be an infinite set of numbers, selected from the natural numbers in a way that looks like the prime number theorem (eg the
n
-th number is approximately
n
log
n
). Then what we've shown is that, if such an
X
obeys the "
X
-Goldbach conjecture" up to
1000
, then we'd expect it to go all the way to infinity.
Thus the Goldbach conjecture can be restated as "in terms of sums of two elements, the prime numbers behave like a typical sequence selected in a prime-number-theorem way".
So the Goldbach conjecture is not saying that there is something special about the primes; in fact, it's saying the opposite, that the primes are typical of similar sequences, that nothing in the specific ways that the primes are selected has an impact on the sum of two primes. So the Goldbach conjecture is essentially saying "there is no obstruction to the primes being typical in this way".
One obstruction
Did you notice that, so far, at no point did I require
N
to be an even number? But all the primes except for
2
are odd. So the distribution of sums of primes is very (very!) heavily skewed towards even numbers; most odd numbers will not appear at all. So, that is one clear obstruction to the possible values of the sum, coming from the way the primes are constructed. The Goldbach conjecture is therefore saying that there are no additional obstructions beyond this one condition on parity.
In fact, the Goldbach conjecture has changed;
1
used to be seen as a prime number, and the original conjecture included
2
1
1
as another example Then
1
was removed from the list of prime numbers, and it turned out, as far as we can tell, that
2
was the only even number we lost from the list of sums.
If we ...
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