|
%I #37 Aug 28 2019 06:43:25
%S 0,2,22,47,422,1047,13547,29172,341672,732297,732297,30029172,
%T 127685422,860107297,4522216672,10625732297,41143310422,498906982297,
%U 1261846435422,5076543701047,62297002685422,348399297607297,1778910772216672,8931468145263547,20852397100341672
%N The successive approximations up to 5^n for 5-adic integer 7^(1/5).
%C For n > 0, a(n) is the unique number k in [1, 5^n] such that k^5 - 7 is divisible by 5^(n+1).
%C For k not divisible by 5, k is a fifth power in 5-adic field if and only if k == 1, 7, 18, 24 (mod 25). If k is a fifth power in 5-adic field, then k has exactly one fifth root.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>
%e For n = 5, we have 1047^5 - 7 = 5^6 * 80521782896, and that 1047 is the unique number k in [1, 5^5] such that k^5 - 7 is divisible by 5^6, so a(5) = 1047.
%o (PARI) a(n) = if(n, lift(sqrtn(7+O(5^(n+1)), 5)), 0)
%Y For the digits of this number see A322169.
%Y For fifth roots in 7-adic field, see A309450, A309451, A309452, A309453, A309454.
%K nonn
%O 0,2
%A _Jianing Song_, Aug 28 2019
|
