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

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, 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, 0, 9, 6, 9, 5, 6, 1, 0, 0, 7, 2, 0, 4, 6, 6, 2, 9, 3, 5, 1

Offset: 0

Author

Seiichi Manyama, Jul 28 2017

Keywords

• nonn
• base

Comments

Also x^2 = A091661.

Examples

			     7^5 -    7 == 0 mod 10,
    57^5 -   57 == 0 mod 10^2,
    57^5 -   57 == 0 mod 10^3,
  7057^5 - 7057 == 0 mod 10^4.
From _Seiichi Manyama_, Aug 01 2019: (Start)
  2^(5^0) + 5^(2^0) ==    7 mod 10,
  2^(5^1) + 5^(2^1) ==   57 mod 10^2,
  2^(5^2) + 5^(2^2) ==   57 mod 10^3,
  2^(5^3) + 5^(2^3) == 7057 mod 10^4. (End)
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..9999

Crossrefs

x^5 = x: A120817 (...6432), A120818 (...3568), A290372 (...5807), A290373 (...2943), this sequence (...7057), A290375 (...4193).

Cf. A091661, A018247.

Programs

Ruby

def P(n)
  s1, s2 = 2, 8
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k1, k2 = j * m + s1, (9 - j) * m + s2
      if (k1 ** 5 - k1) % (m * 10) == 0 && (k2 ** 5 - k2) % (m * 10) == 0
        s1, s2 = k1, k2
        break
      end
    }
  }
  s1
end
def Q(s, n)
  n.times{|i|
    m = 10 ** (i + 1)
    (0..9).each{|j|
      k = j * m + s
      if (k ** 2 - k) % (m * 10) == 0
        s = k
        break
      end
    }
  }
  s
end
def A290374(n)
  str = (P(n) + Q(5, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A290374(100)

Formula

p = A120817 = ...186432, q = A018247 = ...890625, x = p + q = ...077057.