A290373 10-adic integer x = ...2943 satisfying x^5 = x.

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

Offset: 0

Author

Seiichi Manyama, Jul 28 2017

Keywords

• nonn
• base

Comments

Also x^2 = A091661.

Examples

			     3^5 -    3 == 0 mod 10,
    43^5 -   43 == 0 mod 10^2,
   943^5 -  943 == 0 mod 10^3,
  2943^5 - 2943 == 0 mod 10^4.
From _Seiichi Manyama_, Aug 01 2019: (Start)
  8^(5^0) - 5^(2^0) ==    3 mod 10,
  8^(5^1) - 5^(2^1) ==   43 mod 10^2,
  8^(5^2) - 5^(2^2) ==  943 mod 10^3,
  8^(5^3) - 5^(2^3) == 2943 mod 10^4. (End)
		

Links

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

Crossrefs

Cf. A120817, A120818, A290372, A290374, A290375.

Cf. A091661, A120818.

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
    }
  }
  s2
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 A290373(n)
  str = (10 ** (n + 1) + P(n) - Q(5, n)).to_s.reverse
  (0..n).map{|i| str[i].to_i}
end
p A290373(100)

Formula

p = A120818 = ...813568, q = A018247 = ...890625, x = p - q = ...922943.