Let $p_k$ be either primes larger than $40$ or equal to $1$. $n,m$ are larger than $0$ and $b$ is either $1$ or $2$. I'm searching solutions for the following equation: $$ 3\cdot 5 p_1 \pm 37^n p_2 =2^{b}\cdot 29^m p_3 $$ Are there any other methods than brute-force trial-and-error to get the set of solutions for such kind of equations?
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2This is a quite bizarre question. In what arcane place of the jungle did you find it? – chubakueno Apr 23 '14 at 22:06
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@chubakueno here: Unitary Farey Sequence Matrices. It's just a representative of a whole bunch of such equations...but +1 for "arcane place of the jungle" – draks ... Apr 23 '14 at 22:18
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This is not a method for searching for solutions, but there is a theorem of Green and Tao which ensures that for any given $b,n,m$, the number of such primes is asymptotically what you would naively expect it to be. When I say "asymptotically", I mean that if you look for solutions with $p_i$ up to some large number $N$, and you ignore lower order terms. The "naive" expectation is what you would get if you started off pretending that each number $n$ is prime with probability $1/\log n$ (that gives you the right density), and then applied corrections to account for correlations between primarity in different congruence classes (e.g. you know that only one even number is prime, so if you know that two numbers have the same parity, then one of them being prime correlates with them being odd, and hence the other one has better chances of being prime as well).
Jakub Konieczny
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