Problem $27\mathrm{.}4.$ Given an untimate$-$debugger that determines if a program halts, show how to resolve each conjecture.

$($b$)$ The twin primes conjecture that $n$ and $n+2$ are prime for infinitely many $n.$

$[$Hint: If necessary, you can use the ultimate$-$debugger on a program which itself uses the ultimate$-$debugger.$]$