Problem 60949. Check the integers additive decomposition conjecture

Problem statement
A conjecture (I rediscovered ?) related to Goldbach's one states that every integer above 2 can be written as the sum of at maximum two prime numbers and the number 1. The goal of this problem is to check this decomposition. Given a positive integer n as an input, your algorithm will return a vector of two primes, [p1, p2], plus potentially the number 1, [1, p1, p2], such that either n = p1 + p2 (case where n is an even number) or n = 1 + p1 + p2 (case where n is an odd number). This p vector will be sorted in ascending order : 1 < p1 < p2 < n. For n = 1 or n = 2 your algorithm should simply return n.
Examples (check the tests below for more)
  • n = 3 => p = [1, 2] ;
  • n = 7 => p = [2, 5] ;
  • n = 17 => p = [1, 3, 13] ;
  • n = 20 => p = [1; 19] ; % p1 may not be prime in this case
  • n = 23 => p = [1, 3, 19] ;
  • n = 60 => p = [1, 59] ; % p1 may not be prime in this case
  • n = 1 => p = 1 ;
  • n = 2 => p = 2;
Tips
Even if maybe not unique, there is always a solution. If you find a case withouit, at least you will have proven the conjecture to be false ! A simple way to start is to begin with seeking p2, the greater prime before n (even when n is prime itself. Then if the difference between n and this number is a prime number, you just have found p1. Else, add 1 and it should complete the sum.
Forbidden functions
  • regexp
  • str2num
  • assignin
  • echo
See also
Solve

Solution Stats

55.56% Correct | 44.44% Incorrect
Last Solution submitted on Aug 17, 2025

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