The Goldbach conjecture is probably correct; so was Fermat’s last theorem

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, , , , 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 be­ing prime, is . So if we sum up all the primes less than , we get differ­ent sums; these sums will be less than .

So, is it­self is one of these sums? Well, the “prob­a­bil­ity” that it’s not the to­tal of any given sum is ; there­fore the prob­a­bil­ity of it be­ing the to­tal of none of the sums is:

So the prob­a­bil­ity of be­ing the to­tal of such a sum is roughly:

There­fore, the prob­a­bil­ity of all num­bers be­ing the to­tal of such a sum is roughly:

Now, the in­finite product con­verges to a non-zero num­ber if and only if the sum con­verges to a finite num­ber. That se­ries can be seen to be con­ver­gent (for ex­am­ple, by not­ing that for large enough and us­ing the com­par­i­son test).

If use com­put­ers to get an es­ti­mate of , 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 . So, if we as­sume it’s valid up to , we nu­mer­i­cally get:

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 be an in­finite set of num­bers, se­lected from the nat­u­ral num­bers in a way that looks like the prime num­ber the­o­rem (eg the -th num­ber is ap­prox­i­mately ). Then what we’ve shown is that, if such an obeys the “-Gold­bach con­jec­ture” up to , 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 to be an even num­ber? But all the primes ex­cept for 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; used to be seen as a prime num­ber, and the origi­nal con­jec­ture in­cluded as an­other ex­am­ple Then was re­moved from the list of prime num­bers, and it turned out, as far as we can tell, that was the only even num­ber we lost from the list of sums.

If we re­moved from the list of primes, we’d only lose as a sum. Similarly, if we strike out the first primes, we ex­pect—on prob­a­bil­is­tic grounds—that “all num­bers greater than a given are the sums of two primes (first primes not in­cluded)”. If that were to fail, then there’s a re­ally in­ter­est­ing ob­struc­tion out there.

Fer­mat’s last the­o­rem was likely (for )

We can show, similarly, that Fer­mat’s last the­o­rem was very likely on prob­a­bil­is­tic grounds. The the­o­rem states that, for , there do not ex­ist nat­u­ral num­bers such that .

Fix and . Count­ing and , there are nat­u­ral num­bers less than or equal to . There­fore there are pos­si­ble , all less than . So the prob­a­bil­ity that any two of these -th pow­ers sum to is .

So the prob­a­bil­ity that there are no ’s such that , is

The sum con­verges. More­over, we can also sum over : . This also con­verges. So the prob­a­bil­ity of Fer­mat’s last the­o­rem was non-zero, at least for ; add on the fact that the the­o­rem was proved for many and checked for many and , means that, even be­fore it was proved, it was very prob­a­ble it was cor­rect.

So An­drew Wiles’s ge­nius was in show­ing there were no un­ex­pected ob­struc­tions for the “likely” out­come to be true. That’s why the proof is so hard: he was try­ing to prove some­thing very “likely”, and show an ab­sence of struc­ture, rather than a pres­ence, with­out know­ing what that struc­ture could be.

Euler’s con­jec­ture was unlikely

Euler’s con­jec­ture was that you needed to sum at least pow­ers of to get an­other power of ; Fer­mat’s last the­o­rem es­tab­lishes this for , and Euler the­o­rised that this ex­tended.

Euler’s the­o­rem is in fact false; for we have three fourth pow­ers that sum to an­other fourth power as:

There are coun­terex­am­ples known for as well, so the con­jec­ture is false, and not just for one value of .

More in­ter­est­ing from our per­spec­tive, we ex­pect it to be false on prob­a­bil­is­tic grounds. Re­call that the ar­gu­ment about Fer­mat’s last the­o­rem does not work for ; it fails be­cause the cru­cial sum is of the type , which di­verges.

Similarly, if we es­ti­mate the prob­a­bil­ity of Euler’s con­jec­ture, we get terms like the fol­low­ing (for some con­stants ):

This goes to zero for the same rea­son as the case.

So, on prob­a­bil­is­tic grounds, we ex­pect Fer­mat’s last the­o­rem to be true for , and we ex­pect Euler’s con­jec­ture to be false.

The only un­ex­pected re­sult here is that Fer­mat’s last the­o­rem and Euler’s con­jec­ture are true for . So some­thing about the struc­ture of the prob­lem for is mov­ing the re­sult away from the prob­a­bil­is­tic out­come.

The “Stu­art con­jec­ture”

Based on what I wrote about Euler’s con­jec­ture, I’ll haz­ard a con­jec­ture my­self, which I be­lieve to be true on prob­a­bil­is­tic grounds. Namely that if there are in­te­gers, whose -th pow­ers sum non-triv­ially to an­other -th power, then is greater than or equal to .

Fer­mat’s last the­o­rem shows this is true for and .