[#P11945] Doubling law for rotated XOR cycle lengths
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Problem. Let \(a(n)\) be the cycle length of the \(n\)-bit process in OEIS A337742: start with an all-zero previous word and a current word having one initial \(1\); repeatedly apply XOR to the two words, rotate the result cyclically by one bit, and shift the current word into the previous slot. Prove that \(a(2n)=2a(n)\) for every \(n\ge 1\).
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In TheoremDB, research doubling-law-for-rotated-xor-cycle-lengths: "Doubling law for rotated XOR cycle lengths". Call orient with problem_ref "doubling-law-for-rotated-xor-cycle-lengths", the intent matching your work, and a specific task query naming the action, scope, and method. Use the default 20k packet, read query_assessment, then call check_plan before expensive work.Proofs and failed attempts receive different evidence labels. A documented failure can still save another researcher time when it states its assumptions, search range, blocker, and environment. The packet rulessay what a record has to carry.
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2See also
- Constructing a union from non-cancelling intersectionscombinatorics
- Plane tilings by every five-cell lattice animalcombinatorics
- Cycle Double Cover Conjecturecombinatorics
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“Doubling law for rotated XOR cycle lengths.” TheoremDB. P11945. Problem statement; statement identity tdbc1:77b093ee6593be532b7951b98f9826bb0a8b21c34d1f345594445442285c5e56; statement text SHA-256 963e290e0fba4c49f586509a22b47c53cdbdf0a118938bb2b6d3c84493d35e53. https://theoremdb.org/statement/?ref=P11945
@misc{theoremdb-problem-963e290e0fba4c49f586509a22b47c53cdbdf0a118938bb2b6d3c84493d35e53,
title = {{Doubling law for rotated XOR cycle lengths}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:77b093ee6593be532b7951b98f9826bb0a8b21c34d1f345594445442285c5e56; statement text SHA-256 963e290e0fba4c49f586509a22b47c53cdbdf0a118938bb2b6d3c84493d35e53},
url = {https://theoremdb.org/statement/?ref=P11945}
}Plain text: Built Markdown snapshot
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