Indices of two consecutive Triangular Num3ers

 
 
The difference of the indices of two consecutive triangular numbers, each a square, is equal to the sum of two consecutive integers the sum of whose squares is a square.

Checking up the first few,
 

T_1 \; = \; 1

T_8 \; = \; 36 \; = \; 6^2
8 \; - \; 1 \; = \; 7 \; = \; 3 \; + \; 4
3^2 \; + \; 4^2 \; = \; 5^2

T_{49} \; = \; 1225 \; = \; 35^2
49 \; - \; 8 \; = \; 41 \; = \; 20 \; + \; 21
20^2 \; + \; 21^2 \; = \; 29^2

T_{288} \; = \; 41616 \; = \; 204^2
288 \; - \; 49 \; = \; 239 \; = \; 119 \; + \; 120
119^2 \; + \; 120^2 \; = \; 169^2

T_{1681} \; = \; 1413721 \; = \; 1189^2
1681 \; - \; 288 \; = \; 1393 \; = \; 696 \; + \; 697
696^2 \; + \; 697^2 \; = \; 985^2

T_{9800} \; = \; 48024900 \; = \; 6930^2
9800 \; - \; 1681 \; = \; 8119 \; = \; 4059 \; + \; 4060
4059^2 \; + \; 4060^2 \; = \; 5741^2

T_{57121} \; = \; 1631432881 \; = \; 40391^2
57121 \; - \; 9800 \; = \; 47321 \; = \; 23660 \; + \; 23661
23660^2 \; + \; 23661^2 \; = \; 33461^2

T_{332928} \; = \; 55420693056 \; = \; 235416^2
332928 \; - \; 57121 \; = \; 275807 \; = \; 137903 \; + \; 137904
137903^2 \; + \; 137904^2 \; = \; 195025^2

T_{1940449} \; = \; 1882672131025 \; = \; 1372105^2
1940449 \; - \; 332928 \; = \; 1607521 \; = \; 803760 \; + \; 803761
803760^2 \; + \; 803761^2 \; = \; 1136689^2

T_{11309768} \; = \; 63955431761796 \; = \; 7997214^2
11309768 \; - \; 1940449 = 9369319 \; = \; 4684659 \; + \; 4684660
4684659^2 \; + \; 4684660^2 \; = \; 6625109^2

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s