Let $p,q_i, i \ge 1$ be primes, $m$ a positive integer. The equation $$ p.\prod_{i=1}^m(q_i-1)-(p-1).\prod_{i=1}^mq_i=2 $$ for $m=1$ has all twin primes $p,q_1=p+2$ as solution.
Are there solutions for all $m\ge2$ ?
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1$\begingroup$ The simplest heuristic suggests probably yes, but it seems unlikely to be possible to prove this. $\endgroup$– Will SawinCommented May 10, 2020 at 17:14
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2$\begingroup$ I'm not sure what to expect. Without the primality condition, we have for each $m$ an equation of the same degree as the number of variables (namely $m+1$), so we expect some power of $\log N$ solutions up to $N$, unless there's a polynomial family. But primality costs a factor of $(\log N)^{m+1}$, suggesting a delicate balance. $\endgroup$– Noam D. ElkiesCommented May 10, 2020 at 17:59
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1$\begingroup$ @NoamD.Elkies I think the leading term cancels, leaving us with a degree $m$ equation. This makes the probable situation more clear. $\endgroup$– Will SawinCommented May 11, 2020 at 1:52
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3$\begingroup$ In the case $m=2$, you can take $q_1=p+2$ and $q_2=(p^2+p+2)/2$, so that your desired equation is just a polynomial identity. Standard conjectures then ensure that $p, q_1$ and $q_2$ are simultaneously prime, infinitely often $\endgroup$– Mike BennettCommented May 11, 2020 at 4:28
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2 Answers
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An exhaustive search up to $1000$ finds the solutions $$ (p,q_1,q_2) = (3, 5, 7), (11, 13, 67), (59, 67, 487), $$ $$ (191, 277, 613), (251, 463, 547), (347, 571, 883) $$ for $m=2$ and $$ (p,q_1,q_2,q_3) = (3, 5, 7, 37), (3, 7, 7, 11), (23, 37, 107, 131), (83, 137, 283, 797) $$ for $m=3$.
answered May 10, 2020 at 16:58
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Given an odd prime $p$ and integer $m \geq 2$, we can set $q_1=p+2$ and then, for each $i$ with $2 \leq i \leq m$, $$ q_i = \frac{ 4+ (p-1) \prod_{j=1}^{i-1} q_j}{2}. $$ One can check that $$ p \cdot \prod_{i=1}^m (q_i-1) - (p-1) \cdot \prod_{i=1}^m q_i =2, $$ and each $q_i$ is a polynomial in $p$, of degree $2^{i-1}$ (with coefficients in $\mathbb{Z}[1/2^{i-1}]$).
It is plausible that this construction should produce infinitely many values of $p$ for which $p$ and each $q_i$ are simultaneously prime. That being said, I may be missing some local obstruction. While this very likely leads to infinitely many examples for $m \in \{ 2, 3, 4 \}$, the smallest with $m=4$ being $$ (p,q_1,q_2,q_3,q_4)=(3,5,7,37,1297), $$ examples with $m=5$ are rather thinner on the ground : the smallest is with $p=512351711$.
