Counting Paired DNA Strands over a Length Range — Alternative¶
Codex (GPT-6) alternative: closed-form geometric sums. The main solution proves the fixed-length count and prefix formula. Only the computation of \(s=G(h)\) changes.
When \(q\not\equiv1\pmod P\) ,
Use binary exponentiation for \(q^h\) and \((q-1)^{P-2}\) . Both prefixes of one query share the inverse, and the values are substituted into the main formula for \(H(n)\) . When \(q\equiv1\pmod P\) , every term is one, so set \(G(h)=h\bmod P\) without attempting to invert zero.
The geometric-series identity and prime-modulus inverse prove the general branch correct. The all-ones sum proves the special branch correct, and subtracting prefixes gives the answer. Reduce \(2K\) modulo the modulus first: the valid input \(K=500000004\) reaches the special branch.
Each query takes \(O(\log(R+1)+\log P)\) time and \(O(1)\) space; the modulus is fixed here. Block merging avoids inverses, while this program uses fewer state variables. Both pass the full data.