Triangle of numbers arising from Gilbreath’s conjecture

The task is to find the triangle of numbers arising from Gilbreath’s conjecture.

Gilbreath’s conjecture:
It is observed that given a sequence of prime numbers, a sequence can be formed by the absolute difference between the ith and (i+1)th term of the given sequence and the given process can be repeated to form a triangle of numbers. This numbers when forms the elements of Gilbreath conjecture triangle.

The Gilbreath triangle is formed as follows:



  • Let us take primes: 2, 3, 5, 7.
  • Now the difference between adjacent primes is: 1, 2, 2.
  • Now the difference between adjacent elements is: 1, 0.
  • Now the difference between adjacent elements is: 1.
  • In this way, the Gilbreath triangle is formed as:
    2 3 5 7
     1 2 2
      1 0
       1
    
  • This triangle will be read anti-diagonally upwards as
    2, 1, 3, 1, 2, 5, 1, 0, 2, 7, 

Examples:

Input: n = 10
Output: 2, 1, 3, 1, 2, 
        5, 1, 0, 2, 7,

Input: n = 15
Output: 2, 1, 3, 1, 2, 
        5, 1, 0, 2, 7,
        1, 2, 2, 4, 11

Approach:

  1. The (n, k) th term of the Gilbreath sequence is given by
    • where n>0,
    • F(0, k) is the kth prime number where n = 0.
  2. Define a recursive function and we can map the (n, k)th term in a map and store them to reduce computation. we will fill the 0th row with primes.
  3. Traverse the Gilbreath triangle anti-diagonally upwards so we will start from n = 0, k = 0, and in each step increase the k and decrease the n if n<0 then we will assign n=k and k = 0, in this way we can traverse the triangle anti-diagonally upwards.
  4. We have filled the 0th row with 100 primes. if we need to find larger terms of the series we can increase the primes.

Below is the implementation of the above approach:

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// C++ code for printing the Triangle of numbers
// arising from Gilbreath's conjecture
  
#include <bits/stdc++.h>
using namespace std;
  
// Check whether the number
// is prime or not
bool is_Prime(int n)
{
    if (n < 2)
        return false;
  
    for (int i = 2; i <= sqrt(n); i++)
        if (n % i == 0)
            return false;
    return true;
}
  
// Set the 0th row of the matrix
// with c primes from 0, 0 to 0, c-1
void set_primes(map<int, map<int, int> >& mp,
                map<int,
                    map<int, int> >& hash,
                int c)
{
    int count = 0;
  
    for (int i = 2; count < c; i++) {
        if (is_Prime(i)) {
            mp[0][count++] = i;
            hash[0][count - 1] = 1;
        }
    }
}
  
// Find the n, k term of matrix of
// Gilbreath's conjecture
int Gilbreath(map<int, map<int, int> >& mp,
              map<int, map<int, int> >& hash,
              int n, int k)
{
    if (hash[n][k] != 0)
        return mp[n][k];
  
    // recursively find
    int ans
        = abs(Gilbreath(mp, hash, n - 1, k + 1)
              - Gilbreath(mp, hash, n - 1, k));
  
    // store the ans
    mp[n][k] = ans;
    return ans;
}
  
// Print first n terms of Gilbreath sequence
// successive absolute differences of primes
// read by antidiagonals upwards.
void solve(int n)
{
    int i = 0, j = 0, count = 0;
  
    // map to store the matrix
    // and hash to check if the
    // element is present or not
    map<int, map<int, int> > mp, hash;
  
    // set the primes of first row
    set_primes(mp, hash, 100);
  
    while (count < n) {
  
        // print the Gilbreath number
        cout << Gilbreath(mp, hash, i, j)
             << ", ";
  
        // increase the count
        count++;
  
        // anti diagonal upwards
        i--;
        j++;
  
        if (i < 0) {
            i = j;
            j = 0;
        }
    }
}
  
// Driver code
int main()
{
    int n = 15;
  
    solve(n);
    return 0;
}

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Output:

2, 1, 3, 1, 2, 5, 1, 0, 2, 7, 1, 2, 2, 4, 11,

Reference: http://oeis.org/A036262



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Second year Department of Information Technology Jadavpur University

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