Russian amateur mathematician Viktor Voevodov put forward a conjecture generalizing the conjecture about twin primes. He suggested (in a slightly different formulation) that for any finite increasing sequence of primes, there are infinitely many sequences of primes with the same distances between neighboring primes as the original one (it is supposed that the number of primes < first prime). The conjecture about twin primes is obtained as a special case, if we proceed, for example, from the sequence 3,5. Perhaps this kind of hypothesis has already been expressed by someone earlier. What is known about it?
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4$\begingroup$ This is approximately the Hardy Littlewood k-tuple conjecture, with the initial sequence of primes slightly formalized by what we now call admissible tuples. Something must be done about initial distributions that don't repeat, such as $2, 3$ --- there will never be another pair of primes with the same distance of $1$. A (conjecturally) equivalent formulation would be that if a set of distances occurs twice, then it occurs infinitely many times. $\endgroup$– davidlowrydudaCommented Nov 2, 2022 at 15:49
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1$\begingroup$ it is supposed also that the number of primes < first prime. $\endgroup$– Vladimir47Commented Nov 3, 2022 at 3:28
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As written, this is hopeless false. $(2,3)$ is an obvious counterexample. Slightly less trivially, $(3,5,7)$ is a counterexample.
One can correct for these, and if one does so, one gets a version of the Dickson conjecture which says essentially that for any $a_1,a_2 \dotsc a_k$ and $b_1, b_2 \dotsc b_k$, there are infinitely many $n$ where $a_i + b_i n$ are prime for all $1 \leq i \leq k$ unless there is an obvious divisibility restriction preventing this from happening. See also the Hardy–Littlewood $k$-tuples conjecture.
See also the this is the Bunyakovsky conjecture which has been generalized to Hypothesis H, and the much stronger Bateman–Horn conjecture which makes a similar statement for polynomials of any degree, but also predicts asymptotics for how common the simultaneously prime values are.
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2$\begingroup$ it is supposed also that the number of primes < first prime $\endgroup$ Commented Nov 3, 2022 at 3:28
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1$\begingroup$ @Vladimir47 This condition is sufficient to salvage the conjecture - one just ought to check that a tuple of primes of length smaller than all elements is admissible, which is straightforward. $\endgroup$– WojowuCommented Nov 11, 2022 at 14:23