WON plate102 | World!OfNumbers Palindromic Triangulars containing only 6's by Brian Wallace (email) [ May 28, 2001 ] Brian Wallace noticed the first three terms of sequence A036523 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 Keithreacted [ 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 For example (to take one of the more interesting solutions given there), the 8925662618878671th 387-gonal number is 666666666666666666666. Mike Keith, Word play, math, music A000102 Prime Curios! Prime Puzzle Wikipedia 102 Le nombre 102
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