Triangular numbers are defined as
are among the simplest figurate numbers (see picture aside).
Gauss proved that every number is the sum of at most 3 triangular numbers.
Ming showed that the only triangular numbers
which are also Fibonacci numbers are 1, 3, 21 and 55.
N.Tzanahis B.M.M. de Weger proved that there are
six which are equal to the product
of 3 consecutive integers: , ,
, , and
The first nontrivial palindromic
with palindromic index
. The largest know such number, found by
P.De Geest, is
There are several interesting formulas involving triangular
numbers, for example:
The sum of the reciprocals of triangular number is 2.
By solving the diophantine equation , it is easy to find infinite triangular numbers which are also square. The first are
1, 36, 1225, 41616,
1413721, 48024900, 1631432881, 55420693056,...
The 3 smallest primes which concatenated with the previous prime
gives a triangular number are 1812341, 624403308264975451 and
48127684695939820823. For this 3rd value we have
Below, the spiral pattern of triangular numbers up to . See the page on prime numbers for an explanation and links to similar pictures.
The first triangular numbers are
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,
136, 153, 171, 190, 210, 231, 253, 276, 300 more terms
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many triangular numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p