In recreational mathematics, a repdigit or sometimes monodigit[1] is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of repeated and digit. Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary).

Repdigits are the representation in base of the number where is the repeated digit and is the number of repetitions. For example, the repdigit 77777 in base 10 is .

A variation of repdigits called Brazilian numbers are numbers that can be written as a repdigit in some base, not allowing the repdigit 11, and not allowing the single-digit numbers (or all numbers will be Brazilian). For example, 27 is a Brazilian number because 27 is the repdigit 33 in base 8, while 9 is not a Brazilian number because its only repdigit representation is 118, not allowed in the definition of Brazilian numbers. The representations of the form 11 are considered trivial and are disallowed in the definition of Brazilian numbers, because all natural numbers n greater than two have the representation 11n − 1.[2] 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).

History

The concept of a repdigit has been studied under that name since at least 1974,[3] and earlier Beiler (1966) called them "monodigit numbers".[1] The Brazilian numbers were introduced later, in 1994, in the 9th Iberoamerican Mathematical Olympiad that took place in Fortaleza at Brazil. The first problem in this competition, proposed by Mexico, was as follows:[4]

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 not Brazilian.

Primes and repunits

Main article: Repunit prime

For a repdigit to be prime, it must be a repunit (i.e. the repeating digit is 1) and have a prime number of digits in its base (except trivial single-digit numbers), since, for example, the repdigit 77777 is divisible by 7, in any base > 7. In particular, as Brazilian repunits do not allow the number of digits to be exactly two, Brazilian primes must have an odd prime number of digits.[5] Having an odd prime number of digits is not enough to guarantee that a repunit is prime; for instance, 21 = 1114 = 3 × 7 and 111 = 11110 = 3 × 37 are not prime. In any given base b, every repunit prime in that base with the exception of 11b (if it is prime) is a Brazilian prime. The smallest Brazilian primes are

7 = 1112, 13 = 1113, 31 = 111112 = 1115, 43 = 1116, 73 = 1118, 127 = 11111112, 157 = 11112, ... (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 the "Brazilian primes constant", is slightly larger than 0.33 (sequence A306759 in the OEIS).[6] This convergence implies that the Brazilian primes form a vanishingly small fraction of all prime numbers. For instance, among the 3.7×1010 prime numbers below 1012, only 8.8×104 are Brazilian.

The decimal repunit primes have the form for the values of n listed in OEISA004023. It has been conjectured that there are infinitely many decimal repunit primes.[7] The binary repunits are the Mersenne numbers and the binary repunit primes are the Mersenne primes.

It is unknown whether there are infinitely many Brazilian primes. If the Bateman–Horn conjecture is true, then for every prime number of digits there would exist infinitely many repunit primes with that number of digits (and consequentially infinitely many Brazilian primes). Alternatively, if there are infinitely many decimal repunit primes, or infinitely many Mersenne primes, then there are infinitely many Brazilian primes.[8] Because a vanishingly small fraction of primes are Brazilian, there are infinitely many non-Brazilian primes, forming the sequence

2, 3, 5, 11, 17, 19, 23, 29, 37, 41, 47, 53, ... (sequence A220627 in the OEIS)

If a Fermat number is prime, it is not Brazilian, but if it is composite, it is Brazilian.[9] Contradicting a previous conjecture,[10] Resta, Marcus, Grantham, and Graves found examples of Sophie Germain primes that are Brazilian, the first one is 28792661 = 1111173.[11]

Non-Brazilian composites and repunit powers

The only positive integers that can be non-Brazilian are 1, 6, the primes, and the squares of the primes, for every other number is the product of two factors x and y with 1 < x < y − 1, and can be written as xx in base y − 1.[12] If a square of a prime p2 is Brazilian, then prime p must satisfy the Diophantine equation

p2 = 1 + b + b2 + ... + bq-1 with p, q ≥ 3 primes and b >= 2.

Norwegian mathematician Trygve Nagell has proved[13] that this equation has only one solution when p is prime corresponding to (p, b, q) = (11, 3, 5). Therefore, the only squared prime that is Brazilian is 112 = 121 = 111113. There is also one more nontrivial repunit square, the solution (p, b, q) = (20, 7, 4) corresponding to 202 = 400 = 11117, but it is not exceptional with respect to the classification of Brazilian numbers because 20 is not prime.

Perfect powers that are repunits with three digits or more in some base b are described by the Diophantine equation of Nagell and Ljunggren[14]

nt = 1 + b + b2 +...+ bq-1 with b, n, t > 1 and q > 2.

Yann Bugeaud and Maurice Mignotte conjecture that only three perfect powers are Brazilian repunits. They are 121, 343, and 400 (sequence A208242 in the OEIS), the two squares listed above and the cube 343 = 73 = 11118.[15]

k-Brazilian numbers

References

  1. ^ a b Beiler, Albert (1966). Recreations in the Theory of Numbers: The Queen of Mathematics Entertains (2 ed.). New York: Dover Publications. p. 83. ISBN 978-0-486-21096-4.
  2. ^ Schott, Bernard (March 2010). "Les nombres brésiliens" (PDF). Quadrature (in French) (76): 30–38. doi:10.1051/quadrature/2010005.
  3. ^ Trigg, Charles W. (1974). "Infinite sequences of palindromic triangular numbers" (PDF). The Fibonacci Quarterly. 12: 209–212. MR 0354535.
  4. ^ Pierre Bornsztein (2001). Hypermath. Paris. Vuibert. p. 7, exercice a35.
  5. ^ Schott (2010), Theorem 2.
  6. ^ Schott (2010), Theorem 4.
  7. ^ Chris Caldwell, "The Prime Glossary: repunit" at The Prime Pages
  8. ^ Schott (2010), Sections V.1 and V.2.
  9. ^ Schott (2010), Proposition 3.
  10. ^ Schott (2010), Conjecture 1.
  11. ^ Grantham, Jon; Graves, Hester (2019). "Brazilian primes which are also Sophie Germain primes". arXiv:1903.04577.
  12. ^ Schott (2010), Theorem 1.
  13. ^ Nagell, Trygve (1921). "Sur l'équation indéterminée (xn-1)/(x-1) = y". Norsk Matematisk Forenings Skrifter. 3 (1): 17–18..
  14. ^ Ljunggren, Wilhelm (1943). "Noen setninger om ubestemte likninger av formen (xn-1)/(x-1) = yq". Norsk Matematisk Tidsskrift (in Norwegian). 25: 17–20..
  15. ^ Bugeaud, Yann; Mignotte, Maurice (2002). "L'équation de Nagell-Ljunggren (xn-1)/(x-1) = yq". L'Enseignement Mathématique. 48: 147–168..
  16. ^ Daniel Lignon (2012). Dictionnaire de (presque) tous les nombres entiers. Paris. Ellipses. p. 420.