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%I #87 May 04 2019 08:44:24
%S 121,343,400
%N Perfect powers y^q with y > 1 and q > 1 which are Brazilian repunits with three or more digits in some base.
%C 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 A) The Nagell-Ljunggren equation has only these three solutions.
%C Considering the current state of our knowledge, this conjecture seems too ambitious, while the next one seems more reasonable.
%C B) The Nagell-Ljunggren equation has only a finite number of solutions.
%C This last conjecture is true if the abc conjecture is true (see article Bugeaud-Mignotte in link (p. 148).
%C Consequence: 121 is the only known square of prime which is Brazilian.
%C There are no other solutions for some base b < 10000.
%C Some theorems and results about this equation:
%C With the exception of the 3 known solutions,
%C 1) for q = 2, there are no other solutions than 11^2 and 20^2,
%C 2) there is no other solution if 3 divides m than 7^3,
%C 3) there is no other solution if 4 divides m than 20^2. - _Bernard Schott_, Apr 29 2019
%C From _David A. Corneth_, Apr 29 2019: (Start)
%C Intersection of A001597 and A053696.
%C a(4) > 10^25 if it exists using constraints above.
%C 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 Y. Bugeaud and M. Mignotte, <a href="http://dx.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 121 = 11^2 = (3^5 - 1)/ (3 - 1) = 11111_3.
%e 343 = 7^3 = (18^3 - 1)/(18 - 1) = 111_18.
%e 400 = 20^2 = (7^4 - 1)/ (7 - 1) = 1111_7.
%o (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 Cf. A001597, A053696, A220571 (Brazilian composites), A307745 (similar but with digits > 1).
%K nonn,base,bref,more
%O 1,1
%A _Bernard Schott_, Jan 11 2013
%E Small edits to the name by _Bernard Schott_, Apr 30 2019
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