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

 

Find other solutions
 

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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.

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