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I was observing the modulo answers to the following expression. Let $p_{n+2}$, $p_{n+1}$ and $p_{n}$ be three consecutive primes; $$ p_{n} p_{n+1} \mod{p_{n+2}} $$

If, we consider a small subset of primes having the following property. $$ p_{n+2} - p_{n+1}=6 $$ $$ p_{n+1} - p_{n}=6 $$

Then, for all primes greater than $59$ satisfying above property, I’ve observed the below to always hold good.

$$ p_{n} p_{n+1} \equiv 72 \mod{p_{n+2}} $$

Although, the number of examples I’ve checked are a few (all primes less than 1600 having this property); I will be thankful to the stack community, if someone could help me with proving or disproving the same.

Martin Sleziak
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Aziz
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    In general, if we have positive integers $x,y,z$ with $y=x+u , z=y+v$ , we have $$xy=(z-v-u)(z-v)\equiv v(u+v)\mod z$$ – Peter Oct 26 '22 at 11:03
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    Why adjective "sexy" ? Is it standard or have you invented it ? – Jean Marie Oct 26 '22 at 11:17
  • @Aziz, I think you can use other words replacing 'sexy' if it is not mentioned in other sources ? – MAS Oct 26 '22 at 11:19
  • @JeanMarie It is a common terminology for consecutive primes with difference $6$ , for $2$ we have "twin-primes" , for $4$ we have "cousin-primes". – Peter Oct 26 '22 at 11:25
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    $\bmod p_{n+2}!:,\ \begin{align}\color{#c00}{p_{n+1}}&\color{#c00}{,\equiv -6}\ p_n&\equiv -12\end{align}!\Rightarrow, \color{#c00}{p_{n+1}}p_n \equiv (\color{#c00}{-6})(-12)\equiv 72,$ by Congruence product Rule – Bill Dubuque Oct 26 '22 at 12:46
  • $!\bmod p_{n+2}!:\ \color{#c00}{p_{n+1}} = p_{n+2}-d\equiv \color{#c00}{-d},,$ so $,p_n = \color{#c00}{p_{n+1}}-c \equiv \color{#c00}{-d}-c,$ so $,p_{n+1}p_n \equiv -d(-d-c)$ by the Congruence Product Rule, slightly more generally. OP is case $c = 6 = d\ \ $ – Bill Dubuque Oct 26 '22 at 13:23

2 Answers2

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We have $$p_np_{n+1}=(p_{n+2}-12)(p_{n+2}-6)=p_{n+2}^2-18p_{n+2}+72\equiv 72\mod p_{n+2}$$

Peter
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If $p_n=2k+1$ then $p_{n+1}=2k+7$ and $p_{n+2}=2k+13$.

Then $p_np_{n+1}=4k^2+16k+7=(2k+13)(2k-5)+72=p_{n+2}(2k-5)+72$.

So, $p_np_{n+1}\equiv72 \pmod {p_{n+2}}$.

I am thinking about $59$. Cities. I think Aziz is from 57.

Bob Dobbs
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