Wednesday, January 05, 2011

Proof by contradiction

Yesterday I asked for the "final term" of a sequence. It may not be immediately obvious that there is for this sequence such a thing, but consider this: A179066 is defined in such a way that term #1124578 must have a digital root of 1 (the digital root of 1124578) and yet be composed of only the digits 0, 3, 6, and 9 (those digits that are not used in 1124578). But numbers made up of only 0s, 3s, 6s, and 9s will be evenly divisible by 3 or, alternatively, leave a remainder of 0, 3, or 6 when divided by 9 (making their digital roots 3, 6, or 9), thus contradicting our already established fact that the number that we are looking for has a digital root of 1. As defined, term #1124578 cannot exist and term #1124577, therefore, is likely the final term of this sequence.

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