Sum of five consecutive cubes is a square

Nick Hobson nickh at
Sat Mar 10 15:29:30 CET 2007

Hi Seqfans,

In the Dover (2005) edition of Dickson's History of the Theory of Numbers,  
Volume 2, Chapter 21, page 587, it is noted: "E. Lucas (275a) stated that  
the sum of the cubes of five consecutive integers is a square only when  
the middle number is 2, 3, 98 or 120."
(275a) Recherches sur l'analyse indeterminee, Moulins, 1873, 92. Extract  
 from Bull. Soc. d'Emulation du Departement de l'Allier, 12, 1873, 532.

The following PARI/GP script finds, additionally, middle number n = 27.   
Is this sequence finite?  Is such a short sequence (six terms, if we  
include n = 0) worth adding to OEIS?

for(n=1, 10^8, if(issquare(5*n*(n*n+6)), print(n)))


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