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10-adic integer x = ...5807 satisfying x^5 = x.
10

%I #37 Aug 01 2019 18:27:28

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

%T 1,1,5,4,5,0,2,2,9,0,6,9,2,1,7,4,7,2,2,2,2,1,7,5,8,7,8,5,2,4,8,0,6,9,

%U 6,4,4,8,5,8,3,0,8,6,5,2,5,0,6,6,9,9,1,5

%N 10-adic integer x = ...5807 satisfying x^5 = x.

%C Also x^2 = A091661.

%H Seiichi Manyama, <a href="/A290372/b290372.txt">Table of n, a(n) for n = 0..9999</a>

%F p = A120817 = ...186432, q = A018247 = ...890625, x = p - q = ...295807.

%e 7^5 - 7 == 0 mod 10,

%e 7^5 - 7 == 0 mod 10^2,

%e 807^5 - 807 == 0 mod 10^3,

%e 5807^5 - 5807 == 0 mod 10^4.

%e From _Seiichi Manyama_, Aug 01 2019: (Start)

%e 2^(5^0) - 5^(2^0) == 7 mod 10,

%e 2^(5^1) - 5^(2^1) == 7 mod 10^2,

%e 2^(5^2) - 5^(2^2) == 807 mod 10^3,

%e 2^(5^3) - 5^(2^3) == 5807 mod 10^4. (End)

%o (Ruby)

%o def P(n)

%o s1, s2 = 2, 8

%o n.times{|i|

%o m = 10 ** (i + 1)

%o (0..9).each{|j|

%o k1, k2 = j * m + s1, (9 - j) * m + s2

%o if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0

%o s1, s2 = k1, k2

%o break

%o end

%o }

%o }

%o s1

%o end

%o def Q(s, n)

%o n.times{|i|

%o m = 10 ** (i + 1)

%o (0..9).each{|j|

%o k = j * m + s

%o if (k ** 2 - k) % (m * 10) == 0

%o s = k

%o break

%o end

%o }

%o }

%o s

%o end

%o def A290372(n)

%o str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse

%o (0..n).map{|i| str[i].to_i}

%o end

%o p A290372(100)

%Y Cf. A120817, A120818, A290373, A290374, A290375.

%Y Cf. A091661, A018247.

%K nonn,base

%O 0,1

%A _Seiichi Manyama_, Jul 28 2017