Editing Repdigit (section)

==Non-Brazilian composites and repunit powers==
The only positive integers that can be non-Brazilian are 1, 6, the [[prime number|prime]]s, and the [[square number|square]]s of the primes, for every other number is the product of two factors ''x'' and ''y'' with 1 < ''x'' < ''y'' &minus; 1, and can be written as ''xx'' in base ''y'' &minus; 1.{{sfnp|Schott|2010|loc=Theorem 1}} If a square of a prime ''p''<sup>2</sup> is Brazilian, then prime ''p'' must satisfy the [[Diophantine equation]] <div style="text-align: center;">''p''<sup>2</sup> = 1 + ''b'' + ''b''<sup>2</sup> + ... + ''b''<sup>''q''-1</sup> with ''p'', ''q'' ≥ 3 primes and ''b'' >= 2.</div>
Norwegian mathematician [[Trygve Nagell]] has proved<ref>{{cite journal|last=Nagell|first=Trygve|title=Sur l'équation indéterminée (x<sup>n</sup>-1)/(x-1) = y |journal=Norsk Matematisk Forenings Skrifter|year=1921|volume=3|issue=1|pages=17–18}}.</ref> that this equation has only one solution when ''p'' is prime corresponding to {{nowrap|1=(''p'', ''b'', ''q'') = (11, 3, 5)}}. Therefore, the only squared prime that is Brazilian is 11<sup>2</sup> = 121 = 11111<sub>3</sub>.
There is also one more nontrivial repunit square, the solution (''p'', ''b'', ''q'') = (20, 7, 4) corresponding to 20<sup>2</sup> = 400 = 1111<sub>7</sub>, but it is not exceptional with respect to the classification of Brazilian numbers because 20 is not prime.

[[Perfect power]]s that are repunits with three digits or more in some base ''b'' are described by the [[Diophantine equation]] of Nagell and [[Wilhelm Ljunggren|Ljunggren]]<ref>{{cite journal|language=no|last=Ljunggren|first=Wilhelm|title=Noen setninger om ubestemte likninger av formen (x<sup>n</sup>-1)/(x-1) = y<sup>q</sup>|journal=Norsk Matematisk Tidsskrift|year=1943|volume=25|pages=17–20}}.</ref><div style="text-align: center;">''n''<sup>''t''</sup> = 1 + ''b'' + ''b''<sup>2</sup> +...+ ''b''<sup>''q''-1</sup> with ''b, n, t'' > 1 and ''q'' > 2.</div>
Yann Bugeaud and Maurice Mignotte conjecture that only three perfect powers are Brazilian repunits. They are 121, 343, and 400 {{OEIS|A208242}}, the two squares listed above and the cube 343 = 7<sup>3</sup> = 111<sub>18</sub>.<ref>{{cite journal|last1=Bugeaud|first1=Yann |last2=Mignotte|first2=Maurice|title=L'équation de Nagell-Ljunggren (x<sup>n</sup>-1)/(x-1) = y<sup>q</sup>|journal=L'Enseignement Mathématique|year=2002|volume=48|pages=147–168|url=https://www.e-periodica.ch/digbib/view?pid=ens-001:2002:48#261}}.</ref>