Testwiki:Reference desk/Archives/Mathematics/2017 May 6

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May 6

Kaprekar numbers

Are Kaprekar numbers idempotent under multiplication modulo 1 less than the next higher power of 10? GeoffreyT2000 (talk) 17:56, 6 May 2017 (UTC)

Apparently, 4879 is a counterexample. It is Kaprekar because $4879^2=23804641$ and $238+04641=4879$, but $4879^2\equiv 7021mod9999$.

Clearly this happens because the split of 23804641 happens at the "wrong" location. I suspect the property should be easily provable for numbers which are Kaprekar by splitting at the middle. -- Meni Rosenfeld (talk) 19:13, 6 May 2017 (UTC)

Among the (base-10) Kaprekar numbers listed in the article, the exceptions are 4879, 5292 and 38962. -- Meni Rosenfeld (talk) 19:19, 6 May 2017 (UTC)