Triangular Numbers!

Number Theory Level 2

A triangular number t n t_n is defined as

t n = t n − 1 + n t_n = t_{n-1} + n , t 1 = 1 t_1 = 1

Is it true that there are infinitely many triangular numbers that are sum of two other triangular numbers?

Yes No

Shanthanu Rai by Shanthanu Rai , 21, India

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2 solutions

Chris Lewis
Jun 13, 2019

Any time n n itself is a triangular number, the identity t n = t n − 1 + n t_n = t_{n-1}+n is an example of a triangular number that is the sum of two other triangular numbers. (eg when n = 6 n=6 , we have 15 + 6 = 21 15+6=21 , and so on)

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Shanthanu Rai
Dec 2, 2018

Yes.

The following identity is true

t x = t y + t z t_x = t_y + t_z

where x = n ( n + 3 ) / 2 + 1 x = {n(n+3)}/2+1 , y = n + 1 y = n+1 and z = n ( n + 3 ) / 2 z = {n(n+3)}/2

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Will you please show how this identity is proved?

Sumant Chopde - 2 years, 4 months ago

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