This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A208242 #91 May 23 2025 02:41:57 %S A208242 121,343,400 %N A208242 Perfect powers y^q with y > 1 and q > 1 which are Brazilian repunits with three or more digits in some base. %C A208242 These three numbers are the only known solutions y^q of the Nagell-Ljunggren equation (b^m-1)/(b-1) = y^q with y > 1, q > 1, b > 1, m > 2. Yann Bugeaud and Maurice Mignotte propose two alternative conjectures: %C A208242 A) The Nagell-Ljunggren equation has only these three solutions. %C A208242 Considering the current state of our knowledge, this conjecture seems too ambitious, while the next one seems more reasonable. %C A208242 B) The Nagell-Ljunggren equation has only a finite number of solutions. %C A208242 This last conjecture is true if the abc conjecture is true (see article Bugeaud-Mignotte in link, p. 148). %C A208242 Consequence: 121 is the only known square of prime which is Brazilian. %C A208242 There are no other solutions for some base b < 10000. %C A208242 Some theorems and results about this equation: %C A208242 With the exception of the 3 known solutions, %C A208242 1) for q = 2, there are no other solutions than 11^2 and 20^2, %C A208242 2) there is no other solution if 3 divides m than 7^3, %C A208242 3) there is no other solution if 4 divides m than 20^2. - _Bernard Schott_, Apr 29 2019 %C A208242 From _David A. Corneth_, Apr 29 2019: (Start) %C A208242 Intersection of A001597 and A053696. %C A208242 a(4) > 10^25 if it exists using constraints above. %C A208242 In the Nagell-Ljunggren equation, we need b > 2. If b = 2, we get y^q = 2^m - 1 which by Catalan's conjecture has no solutions (see A001597). (End) %H A208242 Y. Bugeaud and M. Mignotte, <a href="https://doi.org/10.5169/seals-66071">L'équation de Nagell-Ljunggren (x^n-1)/(x-1) = y^q</a>, Enseign. Math. 48(2002), 147-168. %e A208242 121 = 11^2 = (3^5 - 1)/ (3 - 1) = 11111_3. %e A208242 343 = 7^3 = (18^3 - 1)/(18 - 1) = 111_18. %e A208242 400 = 20^2 = (7^4 - 1)/ (7 - 1) = 1111_7. %o A208242 (PARI) is(n) = if(!ispower(n), return(0)); for(b=2, n-1, my(d=digits(n, b)); if(#d > 2 && vecmin(d)==1 && vecmax(d)==1, return(1))); 0 \\ _Felix Fröhlich_, Apr 29 2019 %Y A208242 Cf. A001597, A053696, A220571 (Brazilian composites), A307745 (similar but with digits > 1). %K A208242 nonn,base,bref,more %O A208242 1,1 %A A208242 _Bernard Schott_, Jan 11 2013 %E A208242 Small edits to the name by _Bernard Schott_, Apr 30 2019