A341328 - OEIS

%I #10 Jan 22 2023 13:14:24

%S 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,

%T 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,

%U 0,7,7,2,1,0,0,3,3,3,9,5,4,8,8,1,4,0,1,2,4,5,2,4

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

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

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

%C 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).

%H M. Vassilev-Missana, <a href="http://nntdm.net/volume-16-2010/number-3/18-24/">Some results on infinite power towers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 16 (2010) No. 3, 18-24.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

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

%t RealDigits[-5*ProductLog[-Log[5]/5]/Log[5], 10, 105]

%t RealDigits[x/.FindRoot[5^x==x^5,{x,1.7},WorkingPrecision->120],10,120][[1]] (* _Harvey P. Dale_, Jan 22 2023 *)

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

%Y Cf. A166928.

%K nonn,cons

%O 1,2

%A _Jianing Song_, Feb 09 2021