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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 T_n=m^3 for some positive integers n and m, where T_n is a triangular number, so

 1/2n(n+1)=m^3.
(1)

But then

 (2n+1)^2-1=(2m)^3
(2)

or

 (2n+1)^2-(2m)^3=1.
(3)

According to Catalan's conjecture (now a theorem), the only pair of perfect powers differing by 1 is 3^2 and 2^3, so the unique cubic triangular number has 2n+1=3 and 2m=2, implying n=m=1.


See also

Cubic Number, Square Triangular Number, Triangular Number

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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. http://www.jaapspies.nl/mathfiles/problem2004-4A.pdf.

Referenced on Wolfram|Alpha

Cubic Triangular Number

Cite this as:

Weisstein, Eric W. "Cubic Triangular Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubicTriangularNumber.html

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