A333828 The 20-adic integer x = ...70D9AE7F1DI8 satisfying x^5 = x.

8, 18, 13, 1, 15, 7, 14, 10, 9, 13, 0, 7, 6, 6, 13, 5, 14, 16, 0, 4, 11, 8, 8, 10, 8, 8, 3, 12, 6, 7, 19, 8, 10, 12, 11, 1, 1, 15, 2, 12, 8, 8, 10, 19, 4, 10, 19, 7, 17, 8, 12, 14, 9, 19, 11, 18, 16, 14, 19, 9, 4, 2, 16, 11, 0, 12, 11, 1, 6, 11, 12, 3, 3, 16, 11

Offset: 0

Author

Patrick A. Thomas, Apr 07 2020

Keywords

• nonn
• base

Comments

Letters A through J represent the base-20 digits 10 through 19, respectively.

Conjecture: There exists a nontrivial n-adic integer x satisfying x^5 = x, and x^2, x^3, and x^4 are not x, if and only if n has a prime factor of the form 4k+1. Further, there is one nontrivial pair (x and -x) for each different prime factor of the form 4k+1.

Examples

			8^25 in base 20 ends in the digits 13, 18, 8 (or ...DI8 in extended hexadecimal notation).
		

Links

• Patrick A. Thomas, Solutions to x^5=x up to base 100

Crossrefs

Cf. A120817, A210850, A331548.

Formula

The last n+1 digits of 8^(5^n) in base 20, for all n.