OFFSET
1,2
COMMENTS
The trivial solutions with x = 0 and x = 1 are not considered here.
Numbers n such that x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 has nontrivial solutions over the integers.
The nontrivial solutions are found by solving Y^2 = X^3 + d(n)*X with d(n) = n^2*(n^2-1)/4 (A006011), Y = n*y and X = n*x + (1/2)*n*(n-1). [Corrected by Derek Orr, Aug 30 2014]
x^3 + (x+1)^3 + ... + (x+n-1)^3 = y^2 can also be written as y^2 = n(x + (n-1)/2)(n(x + (n-1)/2) + x(x-1)). - Vladimir Pletser, Aug 30 2014
LINKS
Michel Marcus, Examples of triples up to n=50
Vladimir Pletser, Triplets (n, x, y) with n,x less than 10^5
Vladimir Pletser, Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares, Research Gate, 2015.
Vladimir Pletser, Fundamental solutions of the Pell equation X^2-(sigma^4-delta^4)Y^2=delta^4 for the first 45 solutions of the sums of consecutive cubed integers equalling integer squares, Research Gate, 2015. See Reference 19.
V. Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015.
R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.
EXAMPLE
See "Examples of triples" link.
PROG
(PARI)
a(n)=for(x=2, 10^7, /* note this limit only generates the terms in the data section */ X = n*x + (1/2)*n*(n-1); d=n^2*(n^2-1)/4; if(issquare(X^3+d*X), return(x)))
n=1; while(n<100, if(a(n), print1(n, ", ")); n++) \\ Derek Orr, Aug 30 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Nov 08 2012
EXTENSIONS
Name changed, a(1) = 1 prepended and a(39)-a(68) from Derek Orr, Aug 30 2014
More terms for 50<n<10^5 and 1<x<10^5 from Vladimir Pletser, Jan 10 2015
STATUS
approved