perplexus.info :: Numbers : Armstrong plus one : Armstwo

Armstrong plus one : Armstwo (Posted on 2021-12-27)

An Armstrong number is a positive integer that equals the sum of M-th powers of their digits when the number is M-digit long.
153 is an Armstrong number, since: 1³+5³+3³=153.
Sloane's A005188 has an article on this, in which inter-alia it is mentioned that the sequence of Armstrong numbers terminates at the 88th term.

An Armstwo number is a base ten, M-digit long positive integer which is equal to the sum of M-th powers of one greater than each of the digits.
For example, if we check for 153, we find that:
2³+6³+4³= 288, which is NOT equal to 153.

Determine the smallest Armstwo number.

**** Heartfelt thanks to Larry for inspiring this puzzle.

Submitted by K Sengupta

Rating: 5.0000 (1 votes)

Solution: (Hide)

141, 251 and 560 are the only Armstwo numbers with #digits < 9.
Accordingly, 141 is the smallest Armstwo number.

For an explanation, refer to the respective solutions submitted by Larry here and Charlie here.

In this location, Paul has proved that there are no M-digit Armstwo numbers, where M ≥ 60 and, that there are atmost a finite number of solutions, when 3 < M < 60.
He has also conjectured that there are NO solutions when M > 3.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Proof of finite number of solutions	Paul	2021-12-27 20:26:53
MATLAB solution	Charlie	2021-12-27 11:31:01
Solution	Larry	2021-12-27 10:35:31

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