Impossibility of Complete Coverage in [n^2, (n+1)^2]: A Simplified Optimal Allocation Approach to Legendre's Conjecture

1. Concrete Example (N = 7) Concrete Example (N = 7): Number of odd blocks in the interval [49, 64]: 7 + 1 = 8 blocks. Deterministic exclusion: The multiple of 7, given by (7+1)^2 - 1 = 63 (7 \times 9), is excluded \to 7 remaining blocks. Usable odd numbers less than 7: 3, 5 (2 in total). Comparison: Only 2 available elements to cover 7 remaining blocks. Result: Even under the optimal allocation model, the available elements are overwhelmingly insufficient (2 < 7). 2. Generalization General Proposition: For any sequence of n consecutive odd blocks, it is impossible for the multiples of primes (or odd numbers) strictly less than n to completely cover all the blocks. (5, 7, 9, 11, 13, 15