Deterministic Smooth Number Cover and Lemoine Obstructions

Key Points

• The research aims to establish a covering theorem for even numbers using smooth numbers and characterize obstructions below a specific threshold.
• Derived a deterministic covering theorem for the interval [1,N] using Y-smooth numbers.
• Identified conditions under which covering fails based on Lemoine representation of N + 1.
• Proposed a conjecture to strengthen Lemoine-type representations with respect to N/3.
• Confirmed that for even N ≥ 8, the interval can be covered by Y-smooth numbers if Y ≥ ⌈N/3⌉.
• Established that failure in covering occurs if N + 1 can be expressed as p + 2q with primes p, q > Y for Y in a specific range.
• Proposed that such representations could be chosen with primes within a polylogarithmic distance of N/3.