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I've carefully looked at the questions on prime and twin prime, but the following question seems not to habe been asked before.

Context: In the paper by Jeffrey F. Gold and Don H. Tucker titled A Characterization of Twin Prime Pairs (published in: Proceedings NCUR V. (1991), Vol. I, pp. 362-366, see http://www.math.utah.edu/~gold/doc/char.pdf) the authors show the following:

  • there are functions $G$ such that $G(2n)\equiv 1\pmod{6}$ and $G(2n+1)\equiv 5\pmod{6}$, e.g. $G(n)=3n+2^{\sin^2(\frac{n\pi}{2})}$.

  • for integers $i,j\geq 0$ define the following: $n_1(i,j):=8+10i+10j+12ij$, $n_2(i,j):=11+10i+14j+12ij$ and $n_3(i,j):=16+14i+14j+12ij$. Then the primes (greater than 3) are the values of $G(n)$ when excluding the all $n$ such that $n=n_k(i,j)$ for some $i,j\geq 0$ and $k=1,2,3$.

Said differently, one can show there are infinitely many primes by showing that there exists infinetely many $n$ that avoid the ranges of $n_k$ ($k=1,2,3$). And since $n_1$ and $n_3$ take only even values, an infinite sequence of odd $n$ avoiding the range of $n_2$ would prove the infinity of primes.

So it's a bit like saying that primes are numbers that are not of the form $ij$ for some $i,j\geq 0$, except they kind of "zoomed in" by discarding the effect of the primes $2$ and $3$.

Question 1: is it easier, or just as difficult, to find a sequence of odd $n$ avoiding the range of $n_2$ than to find an infinite sequence of $n$ avoiding the range of $n_0(i,j):=ij$ ?

As for twin primes, defining $n_4(i,j):=7+10i+10j+12ij$ and $n_5(i,j):=15+14i+14j+12ij$ they show that twin primes are exactly the pairs $\{G(n);G(n+1)\}$ where $n$ is odd and also avoids the ranges of $n_2$, $n_4$ and $n_5$ (in particular we see that it's a more difficult thing since the values of $n_2$, $n_4$ and $n_5$ are all already odd).

Question 2: here too I'm having trouble to see if that's a step forward or not. The ranges of the $n_k$ are slightly less dense than the original $n_0$, but is that enough to have a grip on things ?

Question 3: as an attempt to find such sequences of $n$, I tried to find some new $m(i,j)$ whose range would be "orthogonal" to that of the $n_k$ but to no avail. Is there some theoretical background on that idea?

Many thanks!

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We could similarly define $m(i,j)=ij+i+j$ as a sieve to cover all the primes, and consider $x^2-1=(s+1)(s^2+2s+t)$ as a sieve for all the twin primes. –  abiessu May 16 at 19:51

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