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Brazilian numbers (Posted on 2017-07-15) Difficulty: 3 of 5
Definition: "Brazilian" numbers ("les nombres brésiliens" in French)are numbers n such that exists a natural number k with 1<k< (n-1) such that the representation of n in base k has all equal digits.

1.Prove that all even numbers above 6 are Brazilian numbers.
2. How many odd Brazilian numbers are there below 100?

See The Solution Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Solution Part 1 Solution and more | Comment 1 of 4
Write the even number as 2*m with m>=3. Then the number can be expressed as 22 in base m-1.

This can be generalized to all composite numbers of at least 6.  Factor the number into f*g with f>1 and g>2.  Then the composite can be written as ff in base g-1.

This leaves prime numbers as the only possible numbers that are not Brazilian.  Brazilian primes must necessarily be repunits in their base, otherwise there would be an obvious factorization using the repeated digit times a repunit.  The first two Brazilian primes are 7 = 111 base 2 and 13 = 111 base 3.

A search on the OEIS for the sequence of non-Brazilian numbers 1,2,3,4,5,11,17,19 finds no matches!

  Posted by Brian Smith on 2017-07-15 09:49:24
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