38 lines
2.4 KiB
Text
38 lines
2.4 KiB
Text
Brazilian numbers are so called as they were first formally presented at the 1994 math Olympiad ''Olimpiada Iberoamericana de Matematica'' in Fortaleza, Brazil.
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Brazilian numbers are defined as:
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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.
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;E.G.:
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:* '''1, 2 & 3''' can not be Brazilian; there is no base '''B''' that satisfies the condition '''1 < B < N-1'''.
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:* '''4''' is not Brazilian; '''4''' in '''base 2''' is '''100'''. The digits are not all the same.
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:* '''5''' is not Brazilian; '''5''' in '''base 2''' is '''101''', in '''base 3''' is '''12'''. There is no representation where the digits are the same.
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:* '''6''' is not Brazilian; '''6''' in '''base 2''' is '''110''', in '''base 3''' is '''20''', in '''base 4''' is '''12'''. There is no representation where the digits are the same.
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:* '''7''' ''is'' Brazilian; '''7''' in '''base 2''' is '''111'''. There is at least one representation where the digits are all the same.
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:* '''8''' ''is'' Brazilian; '''8''' in '''base 3''' is '''22'''. There is at least one representation where the digits are all the same.
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:* ''and so on...''
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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'''.<BR>
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More common: for all all integers '''R''' and '''S''', where '''R > 1''' and also '''S-1 > R''', then '''R*S''' is Brazilian because '''R*S = R(S-1) + R''', which is '''RR''' in '''base S-1'''<BR>
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The only problematic numbers are squares of primes, where R = S. Only 11^2 is brazilian to base 3.<BR>
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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<BR>
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;Task:
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Write a routine (function, whatever) to determine if a number is Brazilian and use the routine to show here, on this page;
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:* the first '''20''' Brazilian numbers;
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:* the first '''20 odd''' Brazilian numbers;
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:* the first '''20 prime''' Brazilian numbers;
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;See also:
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:* '''[[oeis:A125134|OEIS:A125134 - Brazilian numbers]]'''
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:* '''[[oeis:A257521|OEIS:A257521 - Odd Brazilian numbers]]'''
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:* '''[[oeis:A085104|OEIS:A085104 - Prime Brazilian numbers]]'''
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<br><br>
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