Minimum squared input size for a fixed left-shift binary-GCD iteration count
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The question is whether for each n≥2, among ordered positive integer inputs u≥v>0 taking exactly n iterations of the Shallit–Sorenson left-shift binary GCD algorithm, u²+v² has a unique minimum at (p[n−1],p[n−2]). Here p[0]=1, p[1]=3, and p[j+2]=p[j+1]+p[j] for even j, or 2p[j+1]+p[j] for odd j.
Why it matters and what counts as progress
Why it matters: It identifies the uniquely smallest inputs producing a fixed number of iterations of an established binary GCD algorithm.
What counts as progress: Prove the unique extremal input for every iteration count under the exact left-shift algorithm, including the alternating recurrence.
How the problem developed
- 1994 · Foundational analysis [1]
Jeffrey O. Shallit and Jonathan P. Sorenson Published an analysis of the left-shift binary GCD algorithm, including its iteration-count behavior; the later minimization question concerns inputs with a prescribed count of those iterations.
Jason Davies, with GPT-6 Astra and Claude Opus 5.5 contributing to the Lean development The pinned Lean project reports a formal proof of the 1994 paper and of its §6 sum-of-squares conjecture, including uniqueness of the minimizing ordered input for each iteration count n ≥ 2.
AI contribution
Sum-of-squares minimizers for a left-shift binary GCD algorithm
For each n≥2, among ordered positive integer inputs u≥v>0 taking exactly n iterations of the Shallit–Sorenson left-shift binary GCD algorithm, u²+v² has a unique minimum at (p[n−1],p[n−2]). Here p[0]=1, p[1]=3, and p[j+2]=p[j+1]+p[j] for even j, or 2p[j+1]+p[j] for odd j.
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Sum-of-squares minimizers for a left-shift binary GCD algorithm