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Hyper-E notation

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E# is not primitive recursive, and specifically the function E(n) = En##n eventually dominates all primitive recursive functions. In fact, in the fast-growing hierarchy, \(n \mapsto E100\#\#n\) dominates \(f_n\) for all \(n < \omega\) and is itself dominated by \(f_\omega\). E# and xE# form part of a larger notation, the Extensible-E System, that also encompasses Cascading-E Notation. Nathan Ho and Wojowu proved termination for the rules of Hyper-E Notation.

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Extended Hyper-E notation

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\omega^{\omega} \omega

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E# is not primitive recursive, and specifically the function E(n) = En##n eventually dominates all primitive recursive functions. In fact, in the fast-growing hierarchy, \(n \mapsto E100\#\#n\) dominates \(f_n\) for all \(n < \omega\) and is itself dominated by \(f_\omega\). E# and xE# form part of a larger notation, the Extensible-E System, that also encompasses Cascading-E Notation. Nathan Ho and Wojowu proved termination for the rules of Hyper-E Notation.