## Consecutive triangular numbers whose products are square numbers

6 terms :

$T_2 \; \times \; T_3 \; \times \; T_4 \; \times \; T_5 \; \times \; T_6 \; \times \; T_7$

$3\times 6\times 10\times 15\times 21\times 28 \; = \; 1260^2$

8 terms :

$T_1 \; \times \; T_2 \; \times \; T_3 \; \times \; T_4 \; \times \; T_5 \; \times \; T_6 \; \times \; T_7 \; \times \; T_8$

$1 \; \times \; 3 \; \times \; 6 \; \times \; 10 \; \times \; 15 \; \times \; 21 \; \times \; 28 \; \times \; 36 \; = \; 7560^2$

10 terms :

$T_8 \; \times \; T_9 \; \times \; T_{10} \; \times \; ... \; \times \; T_{16} \; \times \; T_{17}$

$36 \; \times \; 45 \; \times \; 55 \; \times \; ... \; \times \; 136 \; \times \; 153 \; = \; 3308104800^2$

12 terms :

$T_{10} \; \times \; T_{11} \; \times \; T_{12} \; \times \; ... \; \times \; T_{14} \; \times \; T_{15}$

$10 \; \times \; 15 \; \times \; 21 \; \times \; ... \; \times \; 105 \; \times \; 120 \; = \; 6810804000^2$

14 terms :

$T_{18} \; \times \; T_{19} \; \times \; T_{20} \; \times \; ... \; \times \; T_{30} \; \times \; T_{31}$

$171 \; \times \; 190 \; \times \; 210 \; \times \; ... \; \times \; 465 \; \times \; 496 \; = \; 240814160266680000^2$

16 terms:

$T_2 \; \times \; T_3 \; \times \; T_4 \; \times \; ... \; \times \; T_{16} \; \times \; T_{17}$

$3 \; \times \; 6 \; \times \; 10 \; \times \; ... \; \times \; 136 \; \times \; 153 \; = \; 4168212048000^2$

$T_9 \; \times \; T_{10} \; \times \; T_{11} \; \times \; ... \; \times \; T_{23} \; \times \; T_{24}$

$45 \; \times \; 55 \; \times \; 66 \; \times \; ... \; \times \; 276 \; \times \; 300 \; = \; 100182976573680000^2$

18 terms :

$T_6 \; \times \; T_7 \; \times \; T_8 \; \times \; ... \; \times \; T_{22} \; \times \; T_{23}$

$21 \; \times \; 28 \; \times \; 36 \; \times \; ... \; \times \; 253 \; \times \; 276 \; = \; 841537003218912000^2$

$T_{32} \; \times \; T_{33} \; \times \; T_{34} ... \; \times \; T_{48} \; \times \; T_{49}$

$528 \; \times \; 561 \; \times \; 595 \; \times \; ... \; \times \; 1176 \; \times \; 1225 \; = \; 180602247845440672130880000^2$

20 terms :

$T_{16} \; \times \; T_{17} \; \times \; T_{18} \; \times \; ... \; \times \; T_{34} \; \times \; T_{35}$

$136 \; \times \; 153 \; \times \; 171 \; \times \; ... \; \times \; 595 \; \times \; 630 \; = \; 11575089018175168886400000^2$

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math grad - Interest: Number theory
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### One Response to Consecutive triangular numbers whose products are square numbers

1. paul says:

Here are the odd length number of terms starting at 7 terms, Format {start T Number, n}

7 terms
{1,1260}
{2,7560}
{7,1081080}
{18,181704600}
{25,1325305800}
{56,232373681820}
{121,40941485046000}
{162,300431255244120}
{343,52998536784979800}
{722,9350711527025068200}
{961,68628371664779706780}
{2016,12108781254494004098760}
{4225,2136480820109547135506100}
{5618,15680514765294645944139000}
{11767,2766682672847641174812764040}
{24642,488155782762470857378809333960}

9 terms
{9,551350800}
{72,3944624055372000}
{441,30544026275171835316800}
{2592,237059823587942554806202596000}

11 terms
{11,439977938400}
{88,104823617532039806400}
{539,27556824235430380201935624000}

13 terms
{13,505974629160000}
{104,4015287392483288225304000}

15 terms
{15,792356269264560000}
{120,209475634666000897333339200000}

17 terms
{1,4168212048000}
{8,601097859442080000}
{17,1621160926915289760000}
{32,5160064224155447775168000}

19 terms
{19,4198806800710600478400000}

21 terms
{3,25246110096567360000}
{6,4922991468830635200000}
{21,13427784148672500329923200000}

Paul.