Triangular Num3ers: T(a)+T(b),T(a)+T(c),T(b)+T(c),T(a)+T(b)+T(c)

 
 
Find positive integers   a, \; b, \; c   such that

T_{a} \; + \; T_{b}
T_{b} \; + \; T_{c}
T_{c} \; + \; T_{a}
T_{a} \; + \; T_{b} \; + \; T_{c}

are all triangular numbers.
 

Here are some solutions:
 

T_{11} \; + \; T_{14} \; = \; 171 \; = \; T_{18}
T_{11} \; + \; T_{14} \; = \; 171 \; = \; T_{18}
T_{14} \; + \; T_{14} \; = \; 210 \; = \; T_{20}
T_{11} \; + \; T_{14} \; + \; T_{14} \; = \; 276 \; = \; T_{23}

T_{230} \; + \; T_{741} \; = \; 301476 \; = \; T_{776}
T_{230} \; + \; T_{870} \; = \; 405450 \; = \; T_{900}
T_{741} \; + \; T_{870} \; = \; 653796 \; = \; T_{1143}
T_{230} \; + \; T_{741} \; + \; T_{870} \; = \; 680361 \; = \; T_{1166}

T_{609} \; + \; T_{779} \; = \; 489555 \; = \; T_{989}
T_{609} \; + \; T_{923} \; = \; 612171 \; = \; T_{1106}
T_{779} \; + \; T_{923} \; = \; 730236 \; = \; T_{1208}
T_{609} \; + \; T_{779} \; + \; T_{923} \; = \; 915981 \; = \; T_{1353}

T_{714} \; + \; T_{798} \; = \; 574056 \; = \; T_{1071}
T_{714} \; + \; T_{989} \; = \; 744810 \; = \; T_{1220}
T_{798} \; + \; T_{989} \; = \; 808356 \; = \; T_{1271}
T_{714} \; + \; T_{798} \; + \; T_{989} \; = \; 1063611 \; = \; T_{1458}

T_{1224} \; + \; T_{1716} \; = \; 2222886 \; = \; T_{2108}
T_{1224} \; + \; T_{3219} \; = \; 5932290 \; = \; T_{3444}
T_{1716} \; + \; T_{3219} \; = \; 6655776 \; = \; T_{3648}
T_{1224} \; + \; T_{1716} \; + \; T_{3219} \; = \; 7405476 \; = \; T_{3848}

 
 

Find other solutions.

Can you find an infinite family of solutions?
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

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