A073226 -id:A073226 - cp's OEIS frontend

A348261 Decimal expansion of the nontrivial number x for which x^Pi = Pi^x.

2, 3, 8, 2, 1, 7, 9, 0, 8, 7, 9, 9, 3, 0, 1, 8, 7, 7, 4, 5, 5, 5, 5, 9, 3, 0, 5, 2, 5, 2, 0, 8, 7, 8, 5, 3, 5, 6, 8, 9, 7, 6, 7, 9, 9, 6, 7, 8, 2, 3, 2, 5, 9, 1, 0, 1, 2, 9, 4, 8, 1, 1, 7, 7, 1, 3, 5, 3, 4, 4, 4, 6, 9, 0, 7, 4, 6, 9, 3, 5, 4, 1, 6, 6, 8, 7, 5, 8, 2, 5, 3, 9, 6, 1, 6, 6, 9, 2, 2, 0, 8, 9, 7, 2, 1, 4

Offset: 1

Timothy L. Tiffin, Oct 08 2021

The x-th root of x equals the Pi-th root of Pi: x^(1/x) = Pi^(1/Pi) = A073238 = 1.43961949584759... .

Like Pi, is x also transcendental?

			2.382179087993018774555593052520878...
x^Pi = Pi^x = 15.28621734783496640312486439999472... .

Cf. A000796 (Pi), A049541 (1/Pi), A073238 (Pi^(1/Pi)), A073226 (e^e, see first comment), A231737.

evalf((t-> -LambertW(-t)/t)(log(Pi)/Pi), 120);  # Alois P. Heinz, Oct 13 2021

{a, b} = NSolve[x^Pi == Pi^x, x, WorkingPrecision -> 300]; a; RealDigits[N[x/.a, 300]][[1]]

Equals -Pi*LambertW(-log(Pi)/Pi)/log(Pi). - Alois P. Heinz, Oct 13 2021

A348359 Decimal expansion of the nontrivial number x for which x^phi = phi^x, where phi is the golden ratio (1+sqrt(5))/2.

Original entry on oeis.org

6, 0, 5, 5, 7, 2, 2, 0, 9, 1, 0, 2, 4, 7, 4, 1, 0, 0, 2, 1, 2, 6, 6, 3, 9, 1, 1, 7, 5, 8, 3, 1, 4, 9, 7, 3, 1, 6, 8, 3, 8, 2, 8, 7, 5, 3, 7, 8, 3, 6, 7, 7, 7, 4, 3, 9, 4, 9, 9, 6, 7, 7, 3, 5, 2, 8, 1, 8, 7, 9, 7, 4, 4, 8, 5, 2, 3, 5, 8, 1, 4, 7, 9, 3, 8, 9, 4, 6, 6, 6, 0, 7, 4, 2, 8, 1, 7, 8, 9, 4, 7, 8, 9, 4, 5, 7

Offset: 1

Timothy L. Tiffin, Oct 14 2021

The x-th root of x equals the phi-th root of phi: x^(1/x) = phi^(1/phi) = A185261 = 1.3463608200348694434247534661858... .

Not surprisingly, x appears to be irrational. If x is also algebraic, then x^phi would be transcendental by the Gelfond-Schneider theorem.

			6.055722091024741002126639117583149731683828...
x^phi = phi^x  = 18.431940924839652158136364051482054378959672... .

Wikipedia, Gelfond-Schneider theorem

Cf. A001622 (phi), A094214 (1/phi), A185261 (phi^(1/phi)), A073226 (e^e, see first comment).

RealDigits[x/.FindRoot[x^GoldenRatio==GoldenRatio^x,{x,6},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Dec 09 2024 *)

Prior Mathematica program replaced by Harvey P. Dale, Dec 09 2024

cp's OEIS Frontend

A348261 Decimal expansion of the nontrivial number x for which x^Pi = Pi^x.

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