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A290374 revision #26

A290374
10-adic integer x = ...7057 satisfying x^5 = x.
10
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
0,1
COMMENTS
Also x^2 = A091661.
LINKS
FORMULA
p = A120817 = ...186432, q = A018247 = ...890625, x = p + q = ...077057.
EXAMPLE
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.
PROG
(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)
m = P(n) + Q(5, n)
m.to_s.reverse.split('').map(&:to_i)[0..n]
end
p A290374(100)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Seiichi Manyama, Jul 28 2017
STATUS
proposed