# Keith Numbers

The number 197 has a curious property:

1 + 9 + 7 = 17
9 + 7 + 17 = 33
7 + 17 + 33 = 57
17 + 33 + 57 = 107
33 + 57 + 107 = 197

After its n digits are used to initiate this pattern, the seeding number itself turns up in the resulting sequence. This makes 197 a Keith number, named for Mike Keith, the mathematician who first remarked on this property in 1987.

Keith numbers are rare and discovered only through exhaustive search, and progress stopped for 13 years after D. Lichtblau found the 34-digit 5752090994058710841670361653731519 in August 2009. But last December, while compiling a programming assignment, Ghent University mathematician Toon Baeyens found all the 35- and 36-digit Keith numbers:

23137274755282109912063039769168142
25314398891465125143523864790391288
44715370344837755402179510861188022
47933465320021485928519060435917729
196866601633638871239614307772203156
214860400509971669129437189647933258
394684240118372710589383926683340073
763701584467955209221750616718219223
880430656963418264331749765271577784

That last entry is now the largest Keith number known.

(Thanks, Peter.)