This is a soft question sparked by my curiosity about the intrinsic depth of Goldbach-like conjectures as perceived by current experts in number theory. The incompleteness theorem implies that, if our chosen foundational system is capable of reasoning about basic arithmetic (equivalently basic string manipulation), then there are true $Π_1$-sentences that we will be unable to prove, not by inability but by impossibility. The thing is, this impossibility arises from the ability to prove finite runs of programs (for some fixed Turing-complete language). But many currently open conjectures are also $Π_1$ too (such as Goldbach's conjecture), and hence also can be interpreted as questions about whether or not certain programs halt.
Based on this, it seems to me that it is very easy to make conjectures about primes that hold up under a statistical assumption on the distribution of primes, at least for sufficiently large numbers, and then tweak the conjecture to eliminate what empirically appears to be the only counter-examples. Just for example, I am no expert in number theory but I can 'randomly' create such a conjecture:
PSQ: Every integer $n>5$ of the form $3k+2$ is the sum of a prime and a positive square.
I checked it using a trivial C program up to $30$ million, and one can see that if we assume an integer $x$ to be a prime with probability $\sim 1/\ln(x)$ then the probability that a number $n$ fails to satisfy PSQ is at most $\sim (1-1/\ln(n/4))^{\sqrt{n}/2}$ $\sim \exp(-\sqrt{n}/\ln(n/4)/2)$ $\ll 1/n^2$, implying that the expected total number of failures is finite.
Under the same probabilistic heuristic, Goldbach's conjecture is even more likely to have finitely many counter-examples than PSQ, but my real questions are not about either of them per se, but rather:
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Should we expect any deep phenomena concerning such conjectures, given that:
- The same probabilistic heuristic applies to other very similar conjectures that have 'random' counter-examples. For example, replacing the "$3k+2$" condition by "non-square" seems (empirically) to give rise to just $38$ counter-examples (the last being $21679$). I hence feel it seems to be a matter of coincidence of the same sort as the law of small numbers, that PSQ is true. (And if it so happens that PSQ is false, we could tweak it as I mentioned earlier, such as requiring $n$ to be a $3k+2$ prime.)
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Even short programs can have complicated behaviour (witness the Busy Beaver function), and if primes are truly distributed 'as randomly as possible', then should we not all the more expect such conjectures about primes to be arising in the same way as coincidental facts concerning long-running programs, namely without any reason?
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I am aware that there may be simple number theoretic constraints. For instance, it makes sense that PSQ has more counter-examples when the $3k+2$ restriction is removed, since on 'average' we expect primes to be equally likely $1$ or $2$ mod $3$, and so 'half the time' the sum of a prime and a square would be $2$ mod $3$. But that merely changes the constants involved in the probabilistic heuristic estimates, and so do not affect my point.
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Are there any uniform explanations (whether theorems or conjectures) that would encompass large classes of such conjectures of sums involving primes? In other words, am I likely wrong in speculating that most such conjectures have coincidental truth values?
Best Answer
There exist surprising counterexamples. Elsholtz and Dietmann found the following: If $p\equiv 7\pmod{8}$ is prime, then the equation $x^2+y^2+z^4=p^2$ has no non-trivial solution. You might argue that this equation is more of Waring then of Goldbach type, but remember that sums of two squares can be described multiplicatively, so it actually is pretty Goldbach like.