When the sum of two Triangular numbers is a perfect cube

 
 
Since   T_n + T_{n-1} = n^2

we may write

T_{n^3} + T_{n^3 - 1}

n^3(n^3 + 1)/2 + (n^3 - 1)(n^3)/2 = (n^2)^3

we find an infinite number of such triangular pairs,

T_1 + T_0 = 1 + 0 = 1^3
T_{2^3} + T_{2^3 - 1} = 36 + 28 = 64 = 4^3
T_{3^3} + T_{3^3 - 1} = 378 + 351 = 729 = 9^3
T_{4^3} + T_{4^3 - 1} = 2080 + 2016 = 4096 = 16^3
T_{5^3} + T_{5^3 - 1} = 7875 + 7750 = 15625 = 25^3
T_{6^3} + T_{6^3 - 1} = 23436 + 23220 = 46656 = 36^3
T_{7^3} + T_{7^3 - 1} = 58996 + 58653 = 117649 = 49^3
T_{8^3} + T_{8^3 - 1} = 131328 + 130816 = 262144 = 64^3
T_{9^3} + T_{9^3 - 1} = 266085 + 265356 = 531441 = 81^3
T_{10^3} + T_{10^3 - 1} = 500500 + 499500 = 1000000 = 100^3
…………
…………..

but not EVERY triangular pair can be found in this manner

 
Other solutions include:
 

Solutions of the form    T_n + T_{2n} :

T_3 + T_6 = 6 + 21 = 27 = 3^3
T_9 + T_{18} = 45 + 171 = 216 = 6^3
T_{48} + T_{96} = 1176 + 4656 = 5832 = 18^3

 

Solutions of the form    T_n + T_{n + k} :

T_3 + T_6 = 27 = 3^3 = (3\times 1)^3

T_9 + T_{18} = 216 = 6^3 = (3\times 2)^3
T_3 + T_{20} = 216 = 6^3 = (3\times 2)^3

T_{26} + T_{27} = 729 = 9^3 = (3\times 3)^3

T_{68} + T_{83} = 5832 = 18^3 = (3\times 6)^3
T_{48} + T_{96} = 5832 = 18^3 = (3\times 6)^3

T_{93} + T_{137} = 13824 = 24^3 = (3\times 8)^3

T_{134} + T_{189} = 27000 = 30^3 = (3\times 10)^3
T_{29} + T_{230} = 27000 = 30^3 = (3\times 10)^3

T_{147} + T_{267} = 46656 = 36^3 = (3\times 12)^3
T_{125} + T_{278} = 46656 = 36^3 = (3\times 12)^3

Note that,
125 = 5^3   and
278 = 125 + 153 = 5^3 + T_{17}

T_{42} + T_{302} = 46656 = 36^3 = (3\times 12)^3

T_{129} + T_{362} = 74088 = 42^3 = (3\times 14)^3

T_{150} + T_{399} = 91125 = 45^3 = (3\times 15)^3
T_{185} + T_{384} = 91125 = 45^3 = (3\times 15)^3

T_{326} + T_{338} = 110592 = 48^3 = (3\times 16)^3
T_{156} + T_{443} = 110592 = 48^3 = (3\times 16)^3

T_{191} + T_{527} = 157464 = 54^3 = (3\times 18)^3

T_{140} + T_{852} = 373248 = 72^3 = (3\times 24)^3

T_{644} + T_{654} = 421875 = 75^3 = (3\times 25)^3
T_{450} + T_{800} = 421875 = 75^3 = (3\times 25)^3

 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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