Brazilian number

This article comes from the article of Wikipédia in French entitled « Nombre brésilien».

Definition and examples

In arithmetic, a Brazilian number is a positive integer of the form

${\displaystyle n\ =\ a{\frac {b^{c}-1}{b-1)),\qquad a<b<n-1,\quad c\geq 2}$.

Hence, a Brazilian number n is such that there is a natural number b with 1 < b < n-1 such that the representation of n in base b has all equal digits. More exactly, n = (aaa...aaa)_b with c times the digit a in a base b.

The condition b < n – 1 is very important because every number n has a representation 11 in base n-1, n = 11_n-1, so every number would be Brazilian.

Examples

27 is a Brazilian number because 27 is the repdigit 33 in base 8: 27 = 33₈.

9 is not a Brazilian number because 9 = 1001₂ = 100₃ = 21₄ = 14₅ = 13₆ = 12₇ and no representation is Brazilian.

History

In 1994, during the 9-th Iberoamerican Mathematical Olympiad that took place in Fortaleza at Brazil, the first problem, proposed by Mexico, has been used by Pierre Bornsztein for his book Hypermath ^[1] : «  A number n > 0 is called « Brazilian » if there exists an integer b such that 1 < b < n – 1 for which the representation of n in base b is written with all equal digits. Prove that 1994 is Brazilian and that 1993 is no Brazilian.»

This exercise was proposed in 2007 on the French mathematical forum les-mathematiques.net ^[2] then Bernard Schott wrote an article about these numbers on the French periodical Quadrature. ^[3]

Some properties

• The smallest Brazilian number is 7 = 111₂ hence 7 is also the smallest Brazilian prime.

• Every even number >= 8 is Brazilian because n = 2 × k = 22_k–1 with k – 1 ≥ 3; so, there exist infinitely many Brazilian numbers and also infinitely many composite Brazilian numbers.

• Every odd number n ≥ 15 equal to n = a × c with a ≥ 3 and c > a + 1 is also Brazilian because n = c × a = (aa)_c–1.

• The first twenty Brazilian numbers are:

7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, … (sequence A125134 in the OEIS).

• The composite Brazilian numbers are: 8, 10, 12, 14, 15, 16, 18, 20, 21, … (sequence A220571 in the OEIS).

• A super-Brazilian number is a repdigit in a base b that is equal to this repdigit. The smallest example is 22₂₂ = 46. The sequence with these particular numbers is OEIS: A287767.

• If n > 7 is no Brazilian, then n is prime or the square of a prime.

• There exist Brazilian primes, Brazilian composites, non-Brazilian primes and non-Brazilian composites. With 0 and 1, these four sets realize a partition of the set ${\displaystyle \mathbb {N} }$ of the natural numbers.

Primes and repunits

Every Brazilian prime is a prime ≥ 7 that is a repunit whose representation has an odd prime number of 1 in a base b, but the reciprocal is false as 21 = 111₄ = 3 × 7 or 111 = 111₁₀ = 3 × 37.

Examples of Brazilian primes : 13 = 111₃ and 127 = 1111111₂.

The Brazilian primes are: 7, 13, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, … (sequence A085104 in the OEIS).

While the sum of the reciprocals of the prime numbers is a divergent series, the sum of the reciprocals of the Brazilian prime numbers is a convergent series whose value called « Brazilian primes constant » is slightly larger than 0.33 (sequence A306759 in the OEIS).

The non-Brazilian primes are: 2, 3, 5, 11, 17, 19, 23, 29, 37, 41, 47, 53, … (sequence A220627 in the OEIS). This series is infinite.

The repunits in decimal written ${\displaystyle R_{n}={\frac {10^{n}-1}{9))\ {\mbox{with ))n\geq 3}$, then ≥ 111 are all Brazilian numbers. The index n of prime Brazilian repunits in base 10 are in OEIS: A004023, except 2 because R₂ = 11 is prime but no Brazilian. It has been conjectured that there are infinitely many repunit primes^[4] and in this case, their number of digits is necessarily prime.

All Mersenne numbers ${\displaystyle M_{n}=2^{n}-1,n\geq 3}$, so ≥ 7 are Brazilian as base-2 repunits. In particular, Mersenne primes >= 7 are Brazilian primes. For example, ${\displaystyle M_{5}=2^{5}-1=31=11111_{2))$.

All Fermat numbers ${\displaystyle F_{n}=2^{2^{n))+1}$ that are primes are no Brazilian while composite Fermat numbers are Brazilian.

The conjecture proposed in Quadrature that no Sophie Germain prime is Brazilian was wrong, Giovanni Resta has showed that the 141385-th Sophie Germain prime 28792661 = 11111₇₃ is Brazilian (comments in sequence OEIS: A085104).

Brazilian primes are relatively rare. There are 16,8% , 7,8% , 5,1% and 3,7% of primes respectively below 10³, 10⁶, 10⁹ and 10¹². Among these primes, there is only 8,3% , 0,26% , 0,0076% and 0,000235% of Brazilian primes. More exactly, below one billion, there are 37 607 912 018 primes but only 88 285 are Brazilian.

However, it is conjectured that there are infinitely many Brazilian primes.

Non-Brazilian composite numbers

Even numbers ≥ 8 and odd numbers with at least two distinct factors are all Brazilian. However, there exist non-Brazilian composites, such as squares of prime numbers 4, 9, 25, 49, but there is only one exception among these squares.

If p² is Brazilian, then prime p must satisfy the Diophantine equation

Norwegian Mathematician Trygve Nagell has proved ^[5] that this equation has only one solution when p is prime corresponding to (p, b, q) = (11, 3, 5) so, 11² = 121 = 11111₃ hence, the only square of prime that is Brazilian is 121.

This equation has another solution when p is composite corresponding to (p, b, q) = (20, 7, 4), with 20² = 400 = 1111₇

When we search perfect powers that are repunits with three digits or more in some base b, we have to solve the Diophantine equation of Nagell-Ljunggren ^[6]

Yann Bugeaud and Maurice Mignotte conjecture ^[7] there are only three perfect powers that are repunits, and so also Brazilian repunits 121, 343 et 400. The new solution is the cube 343 = 7³ = 111₁₈.

There are infinitely many non-Brazilian composites such as the squares of primes ≥ 13. The sequence of the non-Brazilian composites begins with 4, 6, 9, 25, 49, 169, 289, 361, 529,... (OEIS: A190300) and the sequence of non-Brazilian squares of primes is OEIS: A326708.

Numbers several times Brazilian

• There exist numbers that are non-Brazilian and others that are Brazilian; among these last integers, some are once Brazilian, others are twice Brazilian, or three times or four,...A number that is k times Brazilian is called k-Brazilian number.

• Except for numbers 1 and 6, non-Brazilian numbers or numbers 0-Brazilian are constituted only with some primes and some squares of primes. The sequence of the non-Brazilian numbers begins with 1, 2, 3, 4, 5, 6, 9, 11, 17, 19, 23, 25, … (sequence A220570 in the OEIS).

• The sequence of 1-Brazilian numbers is composed of other primes, the only square of prime that is Brazilian, 121, and composite numbers ≥ 8 that are the product of only two distinct factors such that n = a × b = aa_b–1 avec 1 < a < b – 1. (sequence A288783 in the OEIS).

• Among the 2-Brazilian numbers (sequence A290015 in the OEIS), there are composites and only two primes: 31 and 8191. Indeed, according to Goormaghtigh conjecture, these two primes are the only solutions of the Diophantine equation: ${\displaystyle p={\frac {x^{m}-1}{x-1))={\frac {y^{n}-1}{y-1))}$ with x, y > 1 and n, m > 2 with :
  • (p, x, y, m, n) = (31, 5, 2, 3, 5) corresponding to 31 = 11111₂ = 111₅, and,
  • (p, x, y, m, n) = (8191, 90, 2, 3, 13) corresponding to 8191 = 1111111111111₂ = 111₉₀, with 11111111111 is the repunit with thirteen digits 1.

• For each sequence of numbers k-Brazilian, there exists a smallest term. The sequence with these smallest integers k times Brazilian begins with 1, 7, 15, 24, 40, 60, 144, 120, 180, 336, 420, 360, ... and are in OEIS: A284758. For instance, 40 is the smallest number 4-Brazilian with 40 = 1111₃ = 55₇ = 44₉ = 22₁₉.

• In the Dictionnaire de (presque) tous les nombres entiers ^[8], Daniel Lignon proposes that an integer is highly Brazilian if it is a positive integer with more Brazilian representations than any smaller positive integer has. This definition comes from the definition of highly composite numbers created by Srinivasa Ramanujan in 1915.

  The first numbers highly Brazilian are 1, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, ... and are in OEIS: A066044 except for integer 3 that is not a Brazilian number. From 360 to 321 253 732 800 (maybe more), there are 80 successive highly composite numbers that are also highly Brazilian numbers, see OEIS: A279930.

[Please, English is not my native language; this article comes from the article I wrote for Wikipédia in French: https://fr.wikipedia.org/wiki/Nombre_br%C3%A9silien ]. I am ready to answer and help you during your review]. Merci.