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Perfect powers y^m with y > 1 and m > 1 which are Brazilian repdigits with three or more digits > 1 in some base.
2

%I #65 Feb 22 2021 04:00:54

%S 1521,1600,2401,2744,6084,17689,61009,244036,294849,1179396,1483524,

%T 2653641,2725801,2989441,4717584,5239521,7371225,9591409,10614564,

%U 11957764,14447601,17397241,18870336,20277009,20958084,23882769,26904969,29484900,38365636,38825361

%N Perfect powers y^m with y > 1 and m > 1 which are Brazilian repdigits with three or more digits > 1 in some base.

%C The terms of this sequence are solutions y^m of the Diophantine equation a * (b^q - 1)/(b-1) = y^m with 1 < a < b, y >= 2, q >= 3, m >= 2. This equation has been studied by Kustaa A. Inkeri in Acta Arithmetica; some terms of this sequence come from his article where the author limits the study of this equation to bases b <= 100.

%C The case a = 1 is clarified in A208242; it corresponds to the Nagell-Ljunggren equation.

%C The sequence has infinitely many terms because the Diophantine equation 3*(x^2+x+1) = y^2 has infinitely many solutions. - _Giovanni Resta_, Apr 26 2019

%C The corresponding solutions (x, y) of this Diophantine equation are (A028231, A341671).

%C The integers y such that y^2 (m = 2) satisfies this equation are in A158235, except 11 and 20 corresponding to a = 1. - _Bernard Schott_, Apr 27 2019

%H Kustaa A. Inkeri, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21117.pdf">On the Diophantine equation a(x^n-1)/(x-1) = y^m</a>, Acta Arithmetica, XXI, 1972.

%H <a href="/index/Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>.

%e 3 * (22^3-1)/(22-1) = 39^2 and (333)_22 = 39^2 = 1521.

%e 58 * (99^4-1)/(99-1) = 7540^2 and (AAAA)_99 = 7540^2 = 56851600 where A is the symbol for 58 in base 99.

%t rupQ[n_, mx_] := Block[{t, x, p}, p = x^2 + x + 1; While[(t = p /. x -> mx) <= n && Reduce[p == n && x >= mx, x, Integers] === False, p = x*p + 1]; t <= n]; repdQ[n_] := AnyTrue[ Rest@ Most@ Divisors@ n, rupQ[n/#, #+1] &]; ex = 2; up = 10^7; L = {}; While[2^ex <= up, L = Union[L, Parallelize@ Select[ Range[2, Floor[ up^(1/ex)] ]^ex, repdQ]]; ex = NextPrime@ ex]; L (* _Giovanni Resta_, Apr 27 2019 *)

%o (PARI) isokb(n) = for(b=2, n-2, d=digits(n, b); if((#d > 2) && (vecmin(d)==vecmax(d)) && (d[1] > 1), return (1))); 0;

%o isok(n) = ispower(n) && isokb(n); \\ _Michel Marcus_, Apr 28 2019

%Y Subsequence of A001597 and of A125134.

%Y Cf. A158235, A208242 (a=1, that is, with repunits).

%Y Cf. A028231, A341671.

%K nonn,base

%O 1,1

%A _Bernard Schott_, Apr 26 2019

%E More terms from _Giovanni Resta_, Apr 26 2019