Cubic Triangular Number
المؤلف:
Spies, J.
المصدر:
"Problem A NAW 5/5 nr. 4." Dec. 4, 2004. https://www.jaapspies.nl/mathfiles/problem2004-4A.pdf.
الجزء والصفحة:
...
16-12-2020
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Cubic Triangular Number
A cubic triangular number is a positive integer that is simultaneously cubic and triangular. Such a number must therefore satisfy 
for some positive integers 
and 
, where 
is a triangular number, so
 |
(1)
|
But then
 |
(2)
|
or
 |
(3)
|
According to Catalan's conjecture (now a theorem), the only pair of perfect powers differing by 1 is 
and 
, so the unique cubic triangular number has 
and 
, implying 
.
REFERENCES:
Alekseyev, M. "Re: Cube Triangular Numbers and Intersections of Sequences." seqfan@ext.jussieu.fr posting, 30 Dec 2006.
Spies, J. "Problem A NAW 5/5 nr. 4." Dec. 4, 2004. https://www.jaapspies.nl/mathfiles/problem2004-4A.pdf.
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