[ September 2, 2001 ]
A magical blending of different concepts
in a most beautiful pattern
Terry Trotter (email) and pdg
We start with palindromes whose digits smoothly go down and up.
These palindromes are then written out as a product of two near repdigits
and unexpectedly their sum reveals the number of the beast in all his shapes.
Finally we halt with the last entry which is a pandigital palindrome !
All these concepts are blended together in the following beautiful finite
pattern from Terry Trotter, quite a beastly menagerie...
9 = 3 * 3 , Sum = 6
989 = 23 * 43 , Sum = 66
98789 = 223 * 443 , Sum = 666
9876789 = 2223 * 4443 , Sum = 6666
987656789 = 22223 * 44443 , Sum = 66666
98765456789 = 222223 * 444443 , Sum = 666666
9876543456789 = 2222223 * 4444443 , Sum = 6666666
987654323456789 = 22222223 * 44444443 , Sum = 66666666
98765432123456789 = 222222223 * 444444443 , Sum = 666666666
9876543210123456789 = 2222222223 * 4444444443 , Sum = 6666666666
In an effort to create an infinite pattern I (pdg) took the above palindromes p
and calculated their pth palindrome (1 is the 1st palindrome).
The 989th palindrome is 89098
The 98789th palindrome is 887909788
The 9876789th palindrome is 8876790976788
The pattern becomes clear now from the second entry on. The pth palindrome
is constructed with 88 at the outer ends and 909 in the middle.
The zones inbetween are identical to the middle zone of p itself (98_767_89) !
The 98_765434567_89th palindrome is
The 98_7654321234567_89th palindrome is
The pattern holds even when the zeros start appearing in the middle.
So we have e.g.
The 98_7654321000001234567_89th palindrome is
There is a Java engine for calculating the above palindromes at
The Nth Positive Integer Palindrome Generator