Prime Triplets

Build a triangle from all positive integers in the following way:
 2  3
 4  5  6
 7  8  9 1011 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 2829 30 31 32 33 34 35 3637 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63 64 65 66
. . .
Each positive integer has up to eight neighbours in the triangle.
A set of three primes is called a prime triplet if one of the three primes has the other two as neighbours in the triangle.
For example, in the second row, the prime numbers $2$ and $3$ are elements of some prime triplet.
If row $8$ is considered, it contains two primes which are elements of some prime triplet, i.e. $29$ and $31$.
If row $9$ is considered, it contains only one prime which is an element of some prime triplet: $37$.
Define $S(n)$ as the sum of the primes in row $n$ which are elements of any prime triplet.
Then $S(8)=60$ and $S(9)=37$.
You are given that $S(10000)=950007619$.
Find $S(5678027) + S(7208785)$.

This problem involves a combination of mathematics concepts like number theory, combinatorics and computer programming concepts like Dynamic Programming and Sieve of Eratosthenes.

In the depiction of the triangle, you can identify a pattern or rule in how the first number of each row is determined. You can notice that the number of integers in row `n` is `n`, and that each row starts with the number after the last number of the previous row. To get the first number of a row `n`, you can add all the integers from `1` to `n-1` and then add one. So, the first number for row `n` is `n*(n-1)/2 + 1`.

With this method of determining the first number in each row, you would then determine which numbers in each row are prime and which two prime numbers in the adjacent rows are its neighbors. With this, you can determine the prime triplets in each row.

The next step is to sieve out the primes. There’s a simple algorithm for this named the Sieve of Eratosthenes. This, combined with the formula for the first number in each row, previous paragraph, can be used to solve the problem.

However, executing these steps manually can be difficult and daunting especially for the values for `n` like `5678027` and `7208785` given in the question. So, it’s best to write a computer program to do this. After implementing this in a programming language like Python or Java, you would be able to provide a specific numerical answer to the problem.

Here’s a Python snippet that encapsulates the general idea:

def sieve(n):
prime = [True for i in range(n+1)]
p = 2
while p * p <= n: if prime[p] == True: for i in range(p * 2, n+1, p): prime[i] = False p += 1 return prime def S(n): triangle = [i*(i-1)//2+1 for i in range(n+1)] primes = sieve(triangle[-1] + n + 1) sum_triplet_primes = 0 for i in range(2, n+1): for j in range(triangle[i-1], triangle[i]): if primes[j] and (primes[j-(i-1)] or primes[j-(i-2)]): sum_triplet_primes += j return sum_triplet_primes print(S(5678027) + S(7208785)) ``` Please note that this is not the optimized solution and might run out of computational resources for large numbers. This is meant to be a starting point, and optimisation techniques like expanding the sieve and dynamic programming can be added to it for better efficiency. The exact optimized solution for this problem would involve a lot of advanced mathematical concepts from number theory and combinatorics and decent amount of programming knowledge.

More Answers:
Squarefree Numbers
Coloured Configurations
$60$-degree Triangle Inscribed Circles

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