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A290375 10-adic integer x = ...4193 satisfying x^5 = x.

%I A290375 #34 Aug 01 2019 18:27:48
%S A290375 3,9,1,4,0,7,3,3,3,8,1,4,6,9,9,2,5,1,8,8,5,7,3,1,2,1,2,6,7,5,8,3,8,4,
%T A290375 8,8,4,5,4,9,7,7,0,9,3,0,7,8,2,5,2,7,7,7,7,8,2,4,1,2,1,4,7,5,1,9,3,0,
%U A290375 3,5,5,1,4,1,6,9,1,3,4,7,4,9,3,3,0,0,8,4
%N A290375 10-adic integer x = ...4193 satisfying x^5 = x.
%C A290375 Also x^2 = A091661.
%H A290375 Seiichi Manyama, <a href="/A290375/b290375.txt">Table of n, a(n) for n = 0..9999</a>
%F A290375 p = A120818 = ...813568, q = A018247 = ...890625, x = p + q = ...704193.
%e A290375      3^5 -    3 == 0 mod 10,
%e A290375     93^5 -   93 == 0 mod 10^2,
%e A290375    193^5 -  193 == 0 mod 10^3,
%e A290375   4193^5 - 4193 == 0 mod 10^4.
%e A290375 From _Seiichi Manyama_, Aug 01 2019: (Start)
%e A290375   8^(5^0) + 5^(2^0) ==    3 mod 10,
%e A290375   8^(5^1) + 5^(2^1) ==   93 mod 10^2,
%e A290375   8^(5^2) + 5^(2^2) ==  193 mod 10^3,
%e A290375   8^(5^3) + 5^(2^3) == 4193 mod 10^4. (End)
%o A290375 (Ruby)
%o A290375 def P(n)
%o A290375   s1, s2 = 2, 8
%o A290375   n.times{|i|
%o A290375     m = 10 ** (i + 1)
%o A290375     (0..9).each{|j|
%o A290375       k1, k2 = j * m + s1, (9 - j) * m + s2
%o A290375       if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
%o A290375         s1, s2 = k1, k2
%o A290375         break
%o A290375       end
%o A290375     }
%o A290375   }
%o A290375   s2
%o A290375 end
%o A290375 def Q(s, n)
%o A290375   n.times{|i|
%o A290375     m = 10 ** (i + 1)
%o A290375     (0..9).each{|j|
%o A290375       k = j * m + s
%o A290375       if (k ** 2 - k) % (m * 10) == 0
%o A290375         s = k
%o A290375         break
%o A290375       end
%o A290375     }
%o A290375   }
%o A290375   s
%o A290375 end
%o A290375 def A290375(n)
%o A290375   str = (P(n) + Q(5, n)).to_s.reverse
%o A290375   (0..n).map{|i| str[i].to_i}
%o A290375 end
%o A290375 p A290375(100)
%Y A290375 x^5 = x: A120817 (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), A290374 (...7057), this sequence (...4193).
%Y A290375 Cf. A091661, A120818.
%K A290375 nonn,base
%O A290375 0,1
%A A290375 _Seiichi Manyama_, Jul 28 2017