Sierpiński Type 6ab ± a ± b Problem Explorer

The Sierpiński Problem (1964)

Are there infinitely many positive integers that cannot be expressed in any of the four forms:

• 6ab + a + b = (3a + 1)(2b + 1) - 1
• 6ab + a - b = (3a + 1)(2b - 1) - 1
• 6ab - a + b = (3a - 1)(2b + 1) - 1
• 6ab - a - b = (3a - 1)(2b - 1) - 1

Status: UNSOLVED. Exactly 78 integers ≤ 1000 are known to be uncovered.

Connection to GCD: When GCD(a,b) divides 6, enhanced coverage patterns emerge.

Optimization: For n = 6ab ± a ± b, if a ≤ b, then a ≤ √(n/6), justifying √n search.