## 13331 expressible as a sum of the squares of three consecutive odd triangular numbers

The odd triangular numbers are given by :   1, 3, 15, 21, 45, 55, 91, 105, 153, 171, 231, 253, 325, 351, 435, 465, 561, 595, 703, 741, 861, 903, 1035, 1081, 1225, 1275, 1431, 1485, 1653, 1711, 1891, 1953, 2145, 2211, 2415, 2485, 2701, 2775, 3003, 3081, 3321, 3403, 3655, 3741, 4005, 4095, 4371, 4465, 4753, 4851, …

Formula :   $(2 \, n - 1) \, (2 \, n - 1 - (-1)^{n})/2$

while the even triangular numbers are :   0, 6, 10, 28, 36, 66, 78, 120, 136, 190, 210, 276, 300, 378, 406, 496, 528, 630, 666, 780, 820, 946, 990, 1128, 1176, 1326, 1378, 1540, 1596, 1770, 1830, 2016, 2080, 2278, 2346, 2556, 2628, 2850, 2926, 3160, 3240, 3486, 3570, 3828, 3916, 4186, 4278, 4560, …

Formula :   $(2 \, n + 1) \, (2 \, n + 1 -(-1)^{n})/2$

$45^2 \; + \; 55^2 \; + \; 91^2 \; = \; 13331$   is a palindromic prime number

$13331$   is a sum of the squares of three consecutive odd triangular numbers.

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

### One Response to 13331 expressible as a sum of the squares of three consecutive odd triangular numbers

1. Bajpai says:

13331 is the only palindromic prime number that is the sum of the squares of three consecutive odd triangular numbers T(n) for n<=100000.
As far as the case-II of 'even' triangular numbers is concerned, no palindrome is found in the range n<=100000, that meets the criterion.