TITLE: Can the twin prime problem be solved with a single use of a halting oracle?
QUESTION [9 upvotes]: It occurred to me that if it were possible to determine whether a given program halts, that could be used to answer the twin primes conjecture
A) Write a program which takes input n and then counts upward until it's found n pairs of twin primes
B) Write a program which for any input n returns true if A halts and false otherwise
C) Write a program which counts upward running B on every n until B returns false
D) If C halts, there are finitely many twin primes, otherwise infinite.
I was wondering if there was a way to do this without nesting halting problems... ie if you only get one chance to ask whether a program halts, is that sufficient to answer the twin primes conjecture

REPLY [2 votes]: Let $g(3)=4$, and let $g(n+1)=g(n)!$ for every integer $n \geq 3$.
For an integer $n \geq 3$, let $\Psi_n$ denote the statement:
if a system $$S \subseteq \{x_i!=x_{i+1}: 1 \leq i \leq n-1\} \cup
\{x_i \cdot x_j=x_{j+1}: 1 \leq i \leq j \leq n-1\}$$ has at most
finitely many solutions in positive integers $x_1,\dots,x_n$,
then each such solution $(x_1,\dots,x_n)$ satisfies $x_1,\dots,x_n \leq g(n)$.
We conjecture that the statements $\Psi_3,\dots,\Psi_{16}$ are true.
The statement $\Psi_{16}$ proves the implication:
if there exists a twin prime greater than $g(14)$,
then there are infinitely many twin primes, please see:
A. Tyszka, A common approach to Brocard's problem, Landau's problem,
and the twin prime problem,
http://arxiv.org/abs/1506.08655v21
That is, assuming the statement $\Psi_{16}$,
a simple single query to $0′$ decides the twin
prime problem.