Palindromic Triangulars containing only 6's

by Brian Wallace (email) [ *May 28, 2001* ]

Brian Wallace noticed the first three terms of sequence A036523

T_{3} = 6

T_{5} = 66

T_{8} = 666

and wonders whether there are any more

triangular numbers containing all 6's.

Are there number theoretical reasons why larger strings

containing only 6's cannot occur ? Exist there a proof ?

The answer might be hidden in the reference given in sequence A045914.

Note also the keywords 'full' and 'fini'...

Brian posted his question first to sci.math and Mike Keith

reacted [* May 20, 2001 *] with the following interesting notes.

Ballew and Weger (J. Rec Math, Vol 8, No. 2, 1975)

proved that the only triangular numbers that consist of

one or more like digits are 1, 3, 6, 55, 66 and 666.

The proof is only three pages long but involves tedious

enumeration of cases so I won't try to summarize it here.

I suspect (though don't remember a proof of this being

published anywhere) that even if you generalize to k-gonal

numbers (that is, ask how many k-gonal numbers, for a fixed

k, consist of all 6's) the number of solutions is finite.

However, I_do_know that if you generalize further, and

consider the set of all k-gonal numbers with all values of n and k,

then there are an infinite number of solutions. This is shown in

On Repdigit Polygonal Numbers

For example (to take one of the more interesting solutions

given there), the 8925662618878671^{th} 387-gonal number is

**666666666666666666666**.

Mike Keith, Word play, math, music