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A175155 Numbers m satisfying m^2 + 1 = x^2 * y^3 for positive integers x and y.

%I A175155 #69 Sep 21 2022 02:59:27
%S A175155 0,682,1268860318,1459639851109444,2360712083917682,
%T A175155 86149711981264908618,4392100110703410665318,
%U A175155 8171493471761113423918890682,15203047261220215902863544865414318,5484296027914919579181500526692857773246,28285239023397517753374058381589688919682,12439333951782387734360136352377558500557329868
%N A175155 Numbers m satisfying m^2 + 1 = x^2 * y^3 for positive integers x and y.
%C A175155 This sequence is infinite. The fundamental solution of m^2 + 1 = x^2 y^3 is (m,x,y) = (682,61,5), which means the Pellian equation m^2 - 125x^2 = -1 has the solution (m,x) = (682,61) = (m(1),x(1)). This Pellian equation admits an infinity of solutions (m(2k+1),x(2k+1)), k=1,2,..., given by the following recursive relation, starting with m(1)=682, x(1)= 61: m(2k+1) + x(2k+1)*sqrt(125) = (m(1) + x(1)*sqrt(125))^(2k+1).
%C A175155 Squares of these terms are in A060355, since both a(n)^2 and a(n)^2 + 1 are powerful (A001694). - _Charles R Greathouse IV_, Nov 16 2012
%C A175155 It appears that y = A077426. - _Robert G. Wilson v_, Nov 16 2012
%C A175155 Also m^2 + 1 is powerful. Other solutions arise from solutions x to x^2 - k^3*y^2 = -1. - _Georgi Guninski_, Nov 17 2012
%C A175155 Although it is believed that the b-file is complete for all terms m < 10^100, the search only looked for y < 100000. - _Robert G. Wilson v_, Nov 17 2012
%D A175155 Albert H. Beiler, "The Pellian" (Chap. 22), Recreations in the Theory of Numbers, 2nd ed. NY: Dover, 1966.
%D A175155 A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
%D A175155 J. M. De Koninck, Ces nombres qui nous fascinent, Ellipses, 2008, p. 108.
%H A175155 Robert G. Wilson v, <a href="/A175155/b175155.txt">Table of n, a(n) for n = 1..292</a>. [This list has not been proved to be complete! - _N. J. A. Sloane_, Nov 17 2012]
%H A175155 E. E. Whitford, <a href="http://name.umdl.umich.edu/ABV2773.0001.001">The Pell equation</a>, New York, 1912.
%H A175155 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pell&#39;s_equation">Pell's equation</a>
%F A175155 m(1)=682, x(1) = 61 and m(2k+1) + x(2k+1)*sqrt(125) = (m(1) + x(1)*sqrt(125))^(2k+1) m(2k+1) = C(2k+1,0) * m(1)^(2k+1) + C(2k+1,2)*m(1)^(2k-1)*x(1)^2 + ...
%e A175155 For m=682, m^2 + 1 = 465125 = 61^2 * 5^3.
%p A175155 C:=array(0..20,0..20):C[1,1]=1: C[2,1]=1: n1:=682:x1:=61:for nn from 1 by 2 to 15 do:s:=0:for i from 2 to 15 do:for j from 1 to i do:C[i,j]:= C[i-1,j] + C[i-1,j-1]: od:od:for n from 1 by 2 to nn+1 do:s:=s + C[nn+1,n] * n1^(nn-n+1)*x1^(n-1)*125^((n-1)/2):od:print (s):od: # _Michel Lagneau_
%p A175155 # 2nd program _R. J. Mathar_, Mar 16 2016:
%p A175155 # print (nonsorted!) all solutions of A175155 up to search limit
%p A175155 with(numtheory):
%p A175155 # upper limit for solutions n
%p A175155 nsearchlim := 10^40 :
%p A175155 A175155y := proc(y::integer)
%p A175155     local disc;
%p A175155     disc := y^3 ;
%p A175155     cfrac(sqrt(disc),periodic,quotients) ;
%p A175155 end proc:
%p A175155 for y from 2 do
%p A175155     if issqrfree(y) then
%p A175155         # find continued fraction for x^2-(y^3=disc)*y^2=-1, sqrt(disc)
%p A175155         cf := A175155y(y) ;
%p A175155         nlen :=  nops(op(2,cf)) ;
%p A175155         if type(nlen,odd) then
%p A175155             # fundamental solution
%p A175155             fuso := numtheory[nthconver](cf,nlen-1) ;
%p A175155             fusolx := numer(fuso) ;
%p A175155             fusoly := denom(fuso) ;
%p A175155             solx := fusolx ;
%p A175155             soly := fusoly ;
%p A175155             while solx <= nsearchlim do
%p A175155                 rhhs := solx^2-y^3*soly^2 ;
%p A175155                 if rhhs = -1 then
%p A175155                     # print("n=",solx,"x=",soly,"y=",y^3) ;
%p A175155                     print(solx) ;
%p A175155                 end if;
%p A175155                 # solutions from fundamental solution
%p A175155                 tempx := fusolx*solx+y^3*fusoly*soly ;
%p A175155                 tempy := fusolx*soly+fusoly*solx ;
%p A175155                 solx := tempx ;
%p A175155                 soly := tempy ;
%p A175155             end do;
%p A175155         end if;
%p A175155     fi;
%p A175155 end do:
%t A175155 nmax = 10^50; ymax = 100; instances = 10; fi[y_] := n /. FindInstance[0 <= n <= nmax && x > 0 && n^2 + 1 == x^2*y^3, {n, x}, Integers, instances]; yy = Select[Range[1, ymax, 2], !IntegerQ[Sqrt[#]] && OddQ[ Length[ ContinuedFraction[Sqrt[#]][[2]]]]&]; Join[{0}, fi /@ yy // Flatten // Union // Most] (* _Jean-François Alcover_, Jul 12 2017 *)
%o A175155 (PARI) is(n)=ispowerful(n^2+1) \\ _Charles R Greathouse IV_, Nov 16 2012
%Y A175155 Pellian equation: A006704, A006702, A006705, A006703, A003153, A001571, A001570.
%K A175155 nonn
%O A175155 1,2
%A A175155 _Michel Lagneau_, Feb 27 2010
%E A175155 Added condition that x and y must be positive. Added missing initial term 0. Added warning that b-file has not been proved to be correct - there could be missing entries. - _N. J. A. Sloane_, Nov 17 2012