## Triangular number – System of equations

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:

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

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

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

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

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

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

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

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

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

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

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

1. Paul says:

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.