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If $p_1<p_2<...<p_n$ are $n$ different prime numbers, and the sum of any three numbers is still a prime number, what is the maximum value of $n$?

I only can find $3$ numbers $3$, $5$, $11$ that the sum is $19$ which is still prime...

[Update] I think it is feasible to consider the remainders. There are three prime numbers, each one divided by 3, the remainder can be 1 or 2. If we add the three prime numbers, the sum remainder shouldn't be 3 or 6, so only 4 or 5. Consider n=4, the reminders must contain two 2 and two 1, because three 2 or three 1 will be divided by 3, the sum isn't prime anymore. If n =5, then we must add one more 1 or 2 then the sum can't be prime anymore.

[Conclusion] Max n =4.

edited Mar 25 '24 at 06:47

asked Mar 25 '24 at 02:47

Judy Happy

3

Hint: If you have 3 $1\pmod{3}$ primes, 3 $2\pmod{3}$ primes, or a $1\pmod{3}$ and a $2\pmod{3}$ prime, what happens? – Joshua Wang Mar 25 '24 at 03:07
@JoshuaWang any three, not just any sum – Empy2 Mar 25 '24 at 03:55
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See $5,7,11,181$ – Gerry Myerson Mar 25 '24 at 04:38
(5,7,11,181) correct. – Judy Happy Mar 25 '24 at 06:48
In the update, you also have to account for the possibility that one of the primes is $3$. – Gerry Myerson Mar 26 '24 at 06:08

If $p_1

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