Triangular number – System of equations

Posted on April 13, 2015 by benvitalis

 

An integer of the form   T_{n} = n(n+1)/2   is called a Triangular number.

 

System of equations:

T_{A} \; + \; T_{B} \; = \; T_{C}
T_{A} \; - \; T_{B} \; = \; T_{D}
 
 

one solution is the pair    T_6, \; T_5

T_6 \; + \; T_5 \; = \; 21 \; + \; 15 \; = \; 36 \; = \; T_7
T_6 \; - \; T_5 \; = \; 21 \; - \; 15 \; = \; 6 \; = \; T_2

 
 

Find other pairs.

 
 
Paul found:

T_{18} \; + \; T_{14} \; = \; 171 \; + \; 105 \; = \; 276 \; = \; T_{23}
T_{18} \; - \; T_{14} \; = \; 171 \; - \; 105 \; = \; 66 \; = \; T_{11}

T_{37} \; + \; T_{27} \; = \; 703 \; + \; 378 \; = \; 1081 \; = \; T_{46}
T_{37} \; - \; T_{27} \; = \; 703 \; - \; 378 \; = \; 325 \; = \; T_{25}

T_{44} \; + \; T_{39} \; = \; 990 \; + \; 780 \; = \; 1770 \; = \; T_{59}
T_{44} \; - \; T_{39} \; = \; 990 \; - \; 780 \; = \; 210 \; = \; T_{20}

T_{86} \; + \; T_{65} \; = \; 3741 \; + \; 2145 \; = \; 5886 \; = \; T_{108}
T_{86} \; - \; T_{65} \; = \; 3741 \; - \; 2145 \; = \; 1596 \; = \; T_{56}

T_{91} \; + \; T_{54} \; = \; 4186 \; + \; 1485 \; = \; 5671 \; = \; T_{106}
T_{91} \; - \; T_{54} \; = \; 4186 \; - \; 1485 \; = \; 2701 \; = \; T_{73}

T_{116} \; + \; T_{104} \; = \; 6786 \; + \; 5460 \; = \; 12246 \; = \; T_{156}
T_{116} \; - \; T_{104} \; = \; 6786 \; - \; 5460 \; = \; 1326 \; = \; T_{51}

T_{132} \; + \; T_{125} \; = \; 8778 \; + \; 7875 \; = \; 16653 \; = \; T_{182}
T_{132} \; - \; T_{125} \; = \; 8778 \; - \; 7875 \; = \; 903 \; = \; T_{42}

T_{247} \; + \; T_{242} \; = \; 30628 \; + \; 29403 \; = \; 60031 \; = \; T_{346}
T_{247} \; - \; T_{242} \; = \; 30628 \; - \; 29403 \; = \; 1225 \; = \; T_{49}

T_{278} \; + \; T_{209} \; = \; 38781 \; + \; 21945 \; = \; 60726 \; = \; T_{348}
T_{278} \; - \; T_{209} \; = \; 38781 \; - \; 21945 \; = \; 16836 \; = \; T_{183}

T_{392} \; + \; T_{374} \; = \; 77028 \; + \; 70125 \; = \; 147153 \; = \; T_{542}
T_{392} \; - \; T_{374} \; = \; 77028 \; - \; 70125 \; = \; 6903 \; = \; T_{117}

 
 
 
 
 
 
 
 
 
 
 
 
 

About benvitalis

math grad - Interest: Number theory
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2 Responses to Triangular number – System of equations

  1. Paul says:
    April 13, 2015 at 9:26 am

    I think you may have mislabelled the resultant T values, 36 is T8 and 6 is T3.
    Here are a few I found, with the first T number <500

    T6 + T5 = 21 + 15 = 36 = T8 &
    T6 – T5 = 21 – 15 = 6 = T3
    T18 + T14 = 171 + 105 = 276 = T23 &
    T18 – T14 = 171 – 105 = 66 = T11
    T37 + T27 = 703 + 378 = 1081 = T46 &
    T37 – T27 = 703 – 378 = 325 = T25
    T44 + T39 = 990 + 780 = 1770 = T59 &
    T44 – T39 = 990 – 780 = 210 = T20
    T86 + T65 = 3741 + 2145 = 5886 = T108 &
    T86 – T65 = 3741 – 2145 = 1596 = T56
    T91 + T54 = 4186 + 1485 = 5671 = T106 &
    T91 – T54 = 4186 – 1485 = 2701 = T73
    T116 + T104 = 6786 + 5460 = 12246 = T156 &
    T116 – T104 = 6786 – 5460 = 1326 = T51
    T132 + T125 = 8778 + 7875 = 16653 = T182 &
    T132 – T125 = 8778 – 7875 = 903 = T42
    T247 + T242 = 30628 + 29403 = 60031 = T346 &
    T247 – T242 = 30628 – 29403 = 1225 = T49
    T278 + T209 = 38781 + 21945 = 60726 = T348 &
    T278 – T209 = 38781 – 21945 = 16836 = T183
    T392 + T374 = 77028 + 70125 = 147153 = T542 &
    T392 – T374 = 77028 – 70125 = 6903 = T117

    Paul.

    Reply

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