A341328 Decimal expansion of the smaller solution (i.e., the solution other than x = 5) to 5^x = x^5.

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

Offset: 1

Author

Jianing Song, Feb 09 2021

Keywords

• nonn
• cons

Comments

Also decimal expansion of the other solution to log(x)/x = log(5)/5.

Also the limit of infinite tetration a^a^...^a, where a = 5^(1/5).

Let b be a rational number > e, then: if b is not of the form b = (1 + 1/s)^(s+1) for some positive integer s, then the other solution to b^x = x^b (or equivalently, log(x)/x = log(b)/b) is transcendental. In particular, if b is a positive integer other than 1, 2 and 4, then the other solution to b^x = x^b is transcendental (Vassilev-Missana, p. 23).

Examples

			If x = 1.7649219145257758827587235909114591014..., then log(x)/x = log(5)/5.
		

Links

• M. Vassilev-Missana, Some results on infinite power towers, Notes on Number Theory and Discrete Mathematics, Vol. 16 (2010) No. 3, 18-24.
• Index entries for transcendental numbers

Crossrefs

Cf. A073084, A166928.

Programs

Mathematica

RealDigits[-5*ProductLog[-Log[5]/5]/Log[5], 10, 105]
RealDigits[x/.FindRoot[5^x==x^5,{x,1.7},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Jan 22 2023 *)

PARI

default(realprecision, 92); solve(x=1, 2, 5^x-x^5)

Formula

Equals -(5/log(5))*W(-log(5)/5), where W is the principal branch of the Lambert W function.