How many different (positive) prime numbers can be given such that the sum of any three of them is also a prime number? (Also provide such primes!)





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Let's classify the prime numbers based on the remainder they give when divided by 3. We can take at most two numbers from each class, because if three prime numbers give the same remainder when divided by 3, then their sum is divisible by 3 (and is greater than 3). If we took one from each class, their sum would also be divisible by 3 (and would be greater than 3). Therefore, we can take numbers from at most two classes, and from each class, we can take at most two numbers. Thus, there can be at most 4 prime numbers.

Four such primes indeed exist, as shown by the examples $7, 11, 13, 23$ or 19, 23, 37, 41.