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A126203 Middle number of a set of 5 consecutive integers whose sum of cubes is a square.

%I A126203 #37 Feb 11 2024 17:08:07
%S A126203 0,2,3,27,98,120
%N A126203 Middle number of a set of 5 consecutive integers whose sum of cubes is a square.
%C A126203 That is, numbers n >= 0 such that 5n(n^2+6) is a square.
%C A126203 According to Dickson, Lucas stated that the only terms are 2, 3, 98 and 120 (missing 27).
%C A126203 The Mordell reference shows that there are only finitely many solutions. - _Allan Wilks_, Mar 10 2007
%C A126203 From _Max Alekseyev_, Mar 10 2007: (Start)
%C A126203 It can be shown that all such numbers n can be obtained from elements that are perfect squares in the following 3 recurrent sequence:
%C A126203 1) x(0)=0, x(1)=4, x(k+1) = 98*x(k) - x(k-1). If x(k) is a square then n = 30*x(k). In particular: for k=0, we have n=30*x(0)=0, for k=1, we have n=30*x(1)=120.
%C A126203 2) x(0)=1, x(1)=49, x(k+1) = 38*x(k) - x(k-1). If x(k) is a square then n = 2*x(k). In particular: for k=1, we have n=2*x(1)=2, for k=2, we have n=2*x(1)=98.
%C A126203 3) x(0)=1, x(1)=9, x(k+1) = 8*x(k) - x(k-1). If x(k) is a square then n = 3*x(k). In particular: for k=1, we have n=3*x(1)=3, for k=2, we have n=3*x(1)=27.
%C A126203 It also follows that for any such n one of n/2, n/3, or n/30 is a perfect square. I have tested 10^5 terms of each of the recurrent sequences above and found no new perfect squares. (End)
%C A126203 From _Warut Roonguthai_, Apr 28 2007: (Start)
%C A126203 The sequence is finite because the number of integral points on an elliptic curve is finite; in this case the curve is m^2 = 5n^3 + 30n. Multiplying the equation by 25 and letting y = 5m and x = 5n, we have y^2 = x^3 + 150x. According to Magma, the integral points on this curve are (x, y) = (0, 0), (10, 50), (15, 75), (24, 132), (135, 1575), (490, 10850), (600, 14700). So the list is complete. (End)
%C A126203 This was also confirmed (using Sage) by _Jaap Spies_, May 27 2007
%C A126203 About the second comment: Lucas, however, did actually include 27 in his original note, which can be seen at the link cited. The mistake appears to have originated with Dickson. - _Matt Westwood_, Mar 05 2022
%D A126203 L. E. Dickson, History of the Theory of Numbers, Volume 2, Chapter 21, page 587.
%D A126203 L. J. Mordell, Diophantine Equations, Ac. Press; see Th. 1, Chap. 27, p. 255.
%H A126203 L. E. Dickson, <a href="https://archive.org/details/historyoftheoryo02dickuoft/page/587/mode/1up?view=theater&amp;q=120">History of the Theory of Numbers</a>, Volume 2, Chapter 21, page 587.
%H A126203 Edouard Lucas, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k215388q/f537.item">Recherches sur l'analyse indéterminée</a>, Bull. Soc. d'Emulation du Département de l'Allier, 12, 1873, 532.
%p A126203 q:= n-> issqr((n^2+6)*5*n):
%p A126203 select(q, [$0..150])[];  # _Alois P. Heinz_, Mar 08 2022
%t A126203 Select[Partition[Range[-5,130],5,1],IntegerQ[Sqrt[Total[#^3]]]&][[All,3]] (* _Harvey P. Dale_, Jan 31 2017 *)
%o A126203 (PARI) for(n=1, 10^8, if(issquare(5*n*(n*n+6)), print(n)))
%Y A126203 Cf. A000290, A027604.
%K A126203 nonn,fini,full
%O A126203 1,2
%A A126203 _Nick Hobson_, Mar 10 2007