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 four or more (Posted on 2010-01-25)
Without evaluating the value of N=5*7^34 prove that at least one digit appears in the number four or more times.

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Evaluate log10(5*7^34) = log10(5) + 34*log10(7).  log10(7)=.845 and log10(5)=.699, then log10(5) + 34*log10(7) = 29.43.  Therefore 5*7^34 has 30 digits.

With 30 digits, either all digits occur exactly 3 times, or some digit occurs at least 4 times with another digit occuring at most two times.  If each digit occurs exactly 3 times, then the number must be a multiple of 9, but the prime factors of 5*7^34 are 5 and 7, which are coprime to 9.  Therefore at least one digit occurs four or more times in the expansion of 5*7^34.

For the record, 5*7^34 =
27058 47801 89760 55834 47983 04245
The digits 0,5,7,8 all occur four or more times.

 Posted by Brian Smith on 2010-01-25 14:53:21

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