Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
    
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 '14 at 19:51

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.