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A290372 10-adic integer x = ...5807 satisfying x^5 = x.

%I A290372 #37 Aug 01 2019 18:27:28
%S A290372 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 A290372 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 A290372 6,4,4,8,5,8,3,0,8,6,5,2,5,0,6,6,9,9,1,5
%N A290372 10-adic integer x = ...5807 satisfying x^5 = x.
%C A290372 Also x^2 = A091661.
%H A290372 Seiichi Manyama, <a href="/A290372/b290372.txt">Table of n, a(n) for n = 0..9999</a>
%F A290372 p = A120817 = ...186432, q = A018247 = ...890625, x = p - q = ...295807.
%e A290372      7^5 -    7 == 0 mod 10,
%e A290372      7^5 -    7 == 0 mod 10^2,
%e A290372    807^5 -  807 == 0 mod 10^3,
%e A290372   5807^5 - 5807 == 0 mod 10^4.
%e A290372 From _Seiichi Manyama_, Aug 01 2019: (Start)
%e A290372   2^(5^0) - 5^(2^0) ==    7 mod 10,
%e A290372   2^(5^1) - 5^(2^1) ==    7 mod 10^2,
%e A290372   2^(5^2) - 5^(2^2) ==  807 mod 10^3,
%e A290372   2^(5^3) - 5^(2^3) == 5807 mod 10^4. (End)
%o A290372 (Ruby)
%o A290372 def P(n)
%o A290372   s1, s2 = 2, 8
%o A290372   n.times{|i|
%o A290372     m = 10 ** (i + 1)
%o A290372     (0..9).each{|j|
%o A290372       k1, k2 = j * m + s1, (9 - j) * m + s2
%o A290372       if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
%o A290372         s1, s2 = k1, k2
%o A290372         break
%o A290372       end
%o A290372     }
%o A290372   }
%o A290372   s1
%o A290372 end
%o A290372 def Q(s, n)
%o A290372   n.times{|i|
%o A290372     m = 10 ** (i + 1)
%o A290372     (0..9).each{|j|
%o A290372       k = j * m + s
%o A290372       if (k ** 2 - k) % (m * 10) == 0
%o A290372         s = k
%o A290372         break
%o A290372       end
%o A290372     }
%o A290372   }
%o A290372   s
%o A290372 end
%o A290372 def A290372(n)
%o A290372   str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse
%o A290372   (0..n).map{|i| str[i].to_i}
%o A290372 end
%o A290372 p A290372(100)
%Y A290372 Cf. A120817, A120818, A290373, A290374, A290375.
%Y A290372 Cf. A091661, A018247.
%K A290372 nonn,base
%O A290372 0,1
%A A290372 _Seiichi Manyama_, Jul 28 2017