Problem Statement

Brazilian numbers are so called as they were first formally presented at the 1994 math Olympiad Olimpiada Iberoamericana de Matematica in Fortaleza, Brazil. Brazilian numbers are defined as: The set of positive integer numbers where each number N has at least one natural number B where 1 < B < N-1 where the representation of N in base B has all equal digits.

All even integers 2P >= 8 are Brazilian because 2P = 2(P-1) + 2, which is 22 in base P-1 when P-1 > 2. That becomes true when P >= 4. More common: for all all integers R and S, where R > 1 and also S-1 > R, then RS is Brazilian because RS = R(S-1) + R, which is RR in base S-1 The only problematic numbers are squares of primes, where R = S. Only 11^2 is brazilian to base 3.
All prime integers, that are brazilian, can only have the digit 1. Otherwise one could factor out the digit, therefore it cannot be a prime number. Mostly in form of 111 to base Integer(sqrt(prime number)). Must be an odd count of 1 to stay odd like primes > 2 Write a routine (function, whatever) to determine if a number is Brazilian and use the routine to show here, on this page;

Let's start with the solution:

#lang racket

(require math/number-theory)

(define (repeat-digit? n base d-must-be-1?)
  (call-with-values
   (λ () (quotient/remainder n base))
   (λ (q d) (and (or (not d-must-be-1?) (= d 1)) 
                 (let loop ((n q))
                   (if (zero? n)
                       d
                       (call-with-values
                        (λ () (quotient/remainder n base))
                        (λ (q r) (and (= d r) (loop q))))))))))

(define (brazilian? n (for-prime? #f))
  (for/first ((b (in-range 2 (sub1 n))) #:when (repeat-digit? n b for-prime?)) b))

(define (prime-brazilian? n)
  (and (prime? n) (brazilian? n #t)))

(module+ main
  (displayln "First 20 Brazilian numbers:")
  (stream->list (stream-take (stream-filter brazilian? (in-naturals)) 20))
  (displayln "First 20 odd Brazilian numbers:")
  (stream->list (stream-take (stream-filter brazilian? (stream-filter odd? (in-naturals))) 20))
  (displayln "First 20 prime Brazilian numbers:")
  (stream->list (stream-take (stream-filter prime-brazilian? (stream-filter odd? (in-naturals))) 20)))