[ May 28, 2001 ]
Palindromic Triangulars containing only 6's
Brian Wallace (email)

T₃ = 6
T₅ = 66
T₈ = 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