| rdfs:comment | - \[a \star 1 = a + 1 \pmod{2^n}\] \[a \star (b \star c) = (a \star b) \star (a \star c)\] The period of the function \(a \mapsto 1 \star a\) is a function of \(n\), which we will denote as \(p(n)\). The first few values of \(p(n)\) are \(1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, \ldots\) (OEIS A098820), a slow-growing function. \(p\) is provably divergent in the system ZFC + "there exists a rank-into-rank cardinal." Unfortunately, the latter axiom is so strong that the consistency of the system is seriously doubted. Since the divergence of \(p\) has not been proven otherwise, it remains an unsolved problem.
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| abstract | - \[a \star 1 = a + 1 \pmod{2^n}\] \[a \star (b \star c) = (a \star b) \star (a \star c)\] The period of the function \(a \mapsto 1 \star a\) is a function of \(n\), which we will denote as \(p(n)\). The first few values of \(p(n)\) are \(1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, \ldots\) (OEIS A098820), a slow-growing function. \(p\) is provably divergent in the system ZFC + "there exists a rank-into-rank cardinal." Unfortunately, the latter axiom is so strong that the consistency of the system is seriously doubted. Since the divergence of \(p\) has not been proven otherwise, it remains an unsolved problem. Let \(q(n)\) be minimal so that \(p(q(n)) \geq 2^n\), the "pseudoinverse" of \(p\). \(q\) is a fast-growing function that is total iff \(p\) is divergent. The first few values of \(q\) are \(0, 2, 3, 5, 9\). The existence of \(q(n)\) for \(n \geq 5\) has not even been confirmed, but under the assumption of the previously mentioned axiom, Randall Dougherty has shown that \(q(n + 1) > f_{\omega+1}(\lfloor \log_3 n floor - 1)\) in a slightly modified version of the fast-growing hierarchy, and \(q(5) > f_9(f_8(f_8(254)))\). Dougherty has expressed pessimism about the complexity of proving better lower bounds, and no upper bounds are known as of yet.
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