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A290374 10-adic integer x = ...7057 satisfying x^5 = x.

%I A290374 #34 Aug 01 2019 18:27:42
%S A290374 7,5,0,7,7,0,2,9,0,8,1,4,3,2,5,9,5,3,6,9,1,7,1,8,7,2,0,7,3,6,9,6,1,3,
%T A290374 3,3,7,3,3,2,8,6,5,5,4,6,7,9,1,6,8,3,2,2,4,3,3,3,1,5,0,2,4,3,0,1,9,2,
%U A290374 0,9,6,9,5,6,1,0,0,7,2,0,4,6,6,2,9,3,5,1
%N A290374 10-adic integer x = ...7057 satisfying x^5 = x.
%C A290374 Also x^2 = A091661.
%H A290374 Seiichi Manyama, <a href="/A290374/b290374.txt">Table of n, a(n) for n = 0..9999</a>
%F A290374 p = A120817 = ...186432, q = A018247 = ...890625, x = p + q = ...077057.
%e A290374      7^5 -    7 == 0 mod 10,
%e A290374     57^5 -   57 == 0 mod 10^2,
%e A290374     57^5 -   57 == 0 mod 10^3,
%e A290374   7057^5 - 7057 == 0 mod 10^4.
%e A290374 From _Seiichi Manyama_, Aug 01 2019: (Start)
%e A290374   2^(5^0) + 5^(2^0) ==    7 mod 10,
%e A290374   2^(5^1) + 5^(2^1) ==   57 mod 10^2,
%e A290374   2^(5^2) + 5^(2^2) ==   57 mod 10^3,
%e A290374   2^(5^3) + 5^(2^3) == 7057 mod 10^4. (End)
%o A290374 (Ruby)
%o A290374 def P(n)
%o A290374   s1, s2 = 2, 8
%o A290374   n.times{|i|
%o A290374     m = 10 ** (i + 1)
%o A290374     (0..9).each{|j|
%o A290374       k1, k2 = j * m + s1, (9 - j) * m + s2
%o A290374       if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
%o A290374         s1, s2 = k1, k2
%o A290374         break
%o A290374       end
%o A290374     }
%o A290374   }
%o A290374   s1
%o A290374 end
%o A290374 def Q(s, n)
%o A290374   n.times{|i|
%o A290374     m = 10 ** (i + 1)
%o A290374     (0..9).each{|j|
%o A290374       k = j * m + s
%o A290374       if (k ** 2 - k) % (m * 10) == 0
%o A290374         s = k
%o A290374         break
%o A290374       end
%o A290374     }
%o A290374   }
%o A290374   s
%o A290374 end
%o A290374 def A290374(n)
%o A290374   str = (P(n) + Q(5, n)).to_s.reverse
%o A290374   (0..n).map{|i| str[i].to_i}
%o A290374 end
%o A290374 p A290374(100)
%Y A290374 x^5 = x: A120817 (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), this sequence (...7057), A290375 (...4193).
%Y A290374 Cf. A091661, A018247.
%K A290374 nonn,base
%O A290374 0,1
%A A290374 _Seiichi Manyama_, Jul 28 2017