2

For most open conjectures in number theory , there are good heuristics that they are true : Examples include the Goldbach conjecture, the Collatz conjecture , the Riemann hypothesis and the twin prime conjecture.

Are there conjectures in number theory which are not disproved, but there is also no good heuristic that they are true ?

Peter
  • 86,576
  • 3
    Maybe you mean "experimental evidence" ? –  Apr 03 '19 at 18:35
  • 1
    May be I think even the results on odd perfect numbers also somewhat belong here. Since it is unknown whether there exists such a number, but there are several results as to the bounds on such numbers – vidyarthi Apr 05 '19 at 07:46
  • @vidyarthi In fact, although many mathematicians are convinced that no odd perfect number exists, apart from several restrictions no convincing heuristic is currently known. – Peter Apr 05 '19 at 09:07

3 Answers3

7

The Hardy-Littlewood Conjectures do not have a "good heuristics" and are known to be contradictory to each other. The first one is known as strong twin prime conjecture, and the second one states that $$ \pi(x+y)\le \pi(x)+\pi(y) $$ for all $x,y\ge 2$.

Dietrich Burde
  • 140,055
  • (+1) The probability that e.g. there exists an $m$ such that $135241 m$ falls within a very narrow range $[1000000000000000, 1000000000000010]$ is essentially close to zero, right? If a proof considers this probability to be a true random variable (even though it can actually be calculated deterministically), must such a proof always remain a heuristic, or is there a way to convert such methods into a formal proof? – James Apr 20 '25 at 06:13
0

How about the infinitude of the number of Fermat primes?

Fermat primes are prime numbers of the form \[ 2^{2^{n}} + 1. \]

They are just too large to check their primality even for moderately small values of $n$.

Grown pains
  • 166
  • In this case, there is a strong heuristic for the set being finite, using the same approach as for things like Goldbach and twin prime as mentioned in the original question. – JoshuaZ Sep 26 '19 at 22:42
0

How about prime gap conjectures ?

My Mentor made at least 2 such conjectures :

https://sites.google.com/site/tommy1729/prime-twins-and-prime-constellations-tommys-conjecture

https://sites.google.com/site/tommy1729/prime-gaps-tommys-prime-gap-conjecture

Now do not get confused. For specific cases of them; for large values this follows from the PNT and prime twin heuristics and such.

But for all small values it does not.

And the generality of the conjectures makes it a statement for "infinite many small integers ". The conjectures can not be decided by a finite test.

There are also conjectures here on this site that I have no good heuristics for. And maybe some of them do not have them ?

In general I think many gap or density conjectures are what you are looking for.

Closed forms are another thing.

Another example to show my point might be this one :

A conjecture on the closeness of twin primes

mick
  • 17,886