Triangular numbers : T_{A} * T_{B} = T_{C^2 + C}

 
 
T_{A} \cdot T_{B} \; = \; T_{C^2 + C}

where   A, \; B \; > \; 1
 
 
T_6 \cdot T_4 \; = \; T_{5^2 - 5} \; = \; T_{4^2 + 4}
21 \times 10 \; = \; 210 \; = \; T_{20}

T_{13} \cdot T_9 \; = \; T_{10^2 - 10} \; = \; T_{9^2 + 9}
91 \times 45 \; = \; 4095 \; = \; T_{90}

 
There are infinite many solutions of the form,
T_{A} \cdot T_{B} \; = \; T_{B^2 + B} \; = \; T_{(B+1)^2 - (B+1)}

T_{78} \cdot T_{55} = T_{55^2 + 55} = T_{56^2 - 56}
T_{457} \cdot T_{323} = T_{323^2 + 323} = T_{324^2 - 324}
T_{2666} \cdot T_{1885} = T_{1885^2 + 1885} = T_{1886^2 - 1886}
T_{15541} \cdot T_{10989} = T_{10989^2 + 10989} = T_{10990^2 - 10990}
T_{90582} \cdot T_{64051} = T_{64051^2 + 64051} = T_{64052^2 - 64052}
T_{527953} \cdot T_{373319} = T_{373319^2 + 373319} = T_{373320^2 - 373320}
T_{3077138} \cdot T_{2175865} = T_{2175865^2 + 2175865} = T_{2175866^2 - 2175866}
T_{17934877} \cdot T_{12681873} = T_{12681873^2 + 12681873} = T_{12681874^2 - 12681874}
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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 Triangular numbers : T_{A} * T_{B} = T_{C^2 + C}

  1. paul says:

    Here are a few more

    T37 . T26 = T(27^2 – 27)
    703 x 351 = 246753 = T702

    T51 . T7 = T(17^2 – 17)
    1326 x 28 = 37128 = T272

    T76 . T2 = T(12^2 – 12)
    2926 x 3 = 8778 = T132

    T78 . T55 = T(56^2 – 56)
    3081 x 1540 = 4744740 = T3080

    Paul.

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