Wherever and whenever the whole numbers are divided into groups of even and odd numbers the result can be called a Diophantine sieve. Moreover, the odd numbers can still divide into composite odds and prime numbers. If the group of odd primes include the number 2 (the one and only even prime) then it forms a complete set of prime numbers. However, an open set of Diophantine sieve would have to exclude the number 1 as the union of two disjointed sets: [0, 1) and (1, ∞). Unlike the sieve of Eratosthenes, no multiples are deleted instead they are rearranged into distinctive real rows and columns which elements can be identified by their singular or multiple sieve indices.

Further investigations allow the following conjecture that infinite number of repeated groups can be form but each group has the invariant element number of 6. These 6 elements are a mixture of evens, composite odds, and prime numbers with twin primes always appear in the same group. The group that occurs once contains 6 numbers: 0, 2, 3, 4, 5, and 7 and there are 4 primes. The group repeated twice contains: 6, 8, 9, 10, 11, and 13. Two are primes. Group repeated 3 times contains: 12, 14, 15, 16, 17, and 19. Two are primes. 4 times: 18, 20, 21, 22, 23, and 25. There is only one prime. 5 times contains: 24, 26, 27, 28, 29, and 31. Two are primes. This continues indefinitely. The first group of 6 without any primes is group 20, followed by group 24, group 31, group 34 etc. Since sieve elements have singular or multiple indices, these can be used to prove Goldbach conjecture as well as proving the existence of a simple algorithm for finding prime factors of large numbers.