triangular square pentagonal numbers?

Question

Triangular numbers: n(n + 1)/2
1, 3, 6, 10, 15, 21, 28, 36 …
Square numbers: n²
1, 4, 9, 16, 25, 36
Pentagonal numbers: n(3 n - 1)/2
1, 5, 12, 22, 35, 51, 70 …

Lots of triangular squares (e.g. 36)
Lots of square pentagonals (e.g.9801, 94109401)

I conjecture there are no triangular square pentagonal numbers. Machine search out to around 10^16 puts a lower bound on such a number (and that took a LONG runtime). Can anyone give me a proof, though? I think this would need some quite deep mathematics and would actually be a significant result, if that's any incentive.

Figurate numbers: http://mathworld.wolfram.com/FigurateNumber.html

gianlino · Accepted Answer

There is a list triangular pentagonals and it increases very fast. 

http://mathworld.wolfram.com/PentagonalTriangularNumber.html

 "1, 210, 40755, 7906276, 1533776805, ... (Sloane's A014979)."

The induction rule is N_k = 194 N_(k-1) - N_(k-2) + 16 with N_0 = 0 and N_1 = 1.

Similarly for triangular squares, the induction rule is

 N_k = 34 N_(k-1) - N_(k-2) + 2 with N_0 = 0 and N_1 = 1.

To show that these 2 sequences don't intersect after 0 or 1 is hard. However according to

http://mathworld.wolfram.com/PentagonalSquareTriangularNumber.html

this has been done.

Anonymous · Answer

Pentagonal Numbers List

Mathsorcerer · Answer

Yes, there is one triangular, square, pentagonal number--1.

Of course, that is the trivial result.  I know you want a different one; I am looking into it at this time.

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triangular square pentagonal numbers?

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