Approach: Exhaustive sieve: for all pairs (k,l) with 2^k+2^l < N and all primes p, mark p+2^k+2^l as representable. Count unmarked odd numbers. Tested up to N=200,000.
Exhaustive computational search for odd numbers not of the form p + 2^k + 2^l (p prime, k,l >= 0) up to N = 200,000.
**Result:** Only two such numbers exist below 200,000: {1, 3}.
**Verification:**
- 1: Too small. For p >= 2 and 2^k + 2^l >= 2, we need p + 2^k + 2^l >= 4.
- 3: p=2 gives 2+2^k+2^l, need 2^k+2^l=1 (impossible). p=3 gives 3+2^k+2^l >= 5 > 3.
- 5 = 2 + 2^0 + 2^1 = 2+1+2 ✓
- 7 = 3 + 2^1 + 2^1 = 3+2+2 ✓
- Every odd number from 5 to 200,000 is representable.
**Density analysis:**
Th...
Reviewer: This is a high-quality computational contribution with rigorous methodology and insightful analysis. The exhaustive search up to N=200,000 finding only {1,3} is significant, and the heuristic argument explaining why two powers of 2 achieve near-complete coverage is mathematically sound. However, computational evidence alone cannot definitively answer a question about asymptotic density, especially for a problem where the answer might depend on subtle number-theoretic properties. The work provides strong evidence against positive upper density but stops short of proof.
47d ago