World!Of
Numbers

WON plate
127 |

[ February 25, 2002 ]
Power of Patterns.
Illustrated by Klaus Brockhaus (email).

The pattern represents the new sequence A068065: Palindromes n
for which there is a unique k such that n = k + reverse(k).
E.g. 10801 = 10800 + 00801
and for no other k we have 10801 = k + reverse(k).
The remarkable gap in the second column of the following
pattern arises because 121 = 47 + 74 but also 121 = 110 + 011.

0
101
10001
1000001
100000001
...
2
ooo
10201
1002001
100020001
...
4
141
10401
1004001
100040001
...
6
161
10601
1006001
100060001
...
8
181
10801
1008001
100080001
...
11
1001
100001
10000001
1000000001
...

Asking if the dots at the bottom of each column
indicate an infinite pattern, or just unexplored terrain,
Klaus responded

"I have not worked out a rigorous proof, that the sequence continues in
the way indicated by the dots, but informal considerations make it clear
that the uniqueness condition is very restrictive (if n = a + b and a is
not palindromic or ends with 0, then n = b + a is a second, different
representation) so some case distinctions concerning the number of
digits will lead to the desired result that the pattern is infinite.
A purely computational exploration of the larger numbers is not
feasible because of the required time."

Sloane sequences
A068065, A068064, A068062, A068061, and A067030.

A000127 Prime Curios! Prime Puzzle
Wikipedia 127 Le nombre 127
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