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Algebra Level 2

Yes or No (or Ambiguous )

An example of a number that is triangular and tetrahedral is 10. Are there more numbers that are triangular and tetrahedral, like 10 (other than 1)?

Note : A tetrahedral number is the sum of triangular numbers (starting from 1) and a triangular number is of the form n ( n + 1 ) 2 \dfrac{n(n+1)}{2}

Yes No Ambiguous

Mohammad Farhat by Mohammad Farhat , 16, Singapore

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1 solution

Mohammad Farhat
Oct 15, 2018

Relevant wiki: Gauss: The Prince of Mathematics

A number that is both tetrahedral and triangular satisfy the binomial coefficient equation:

T n = ( n + 1 2 ) = ( m + 2 3 ) = T e m T_n = \dbinom{n+1}{2} = \dbinom{m+2}{3} = Te_m

The only solutions of which are :

T e 3 = T 4 = 10 Te_3 = T_4 =10

T e 8 = T 15 = 120 Te_8 = T_{15} =120

T e 20 = T 55 = 1540 Te_{20} = T_{55} = 1540

T e 34 = T 119 = 7140 Te_{34} = T_{119} = 7140

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