Triangular Number - Other Properties

Other Properties

Triangular numbers correspond to the first-order case of Faulhaber's formula.

Every even perfect number is triangular, given by the formula

where M_p is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are triangular.

For example, the 3rd triangular number is 3x2 = 6; the 7th is 7x4 = 28; the 31st is 31x16 = 496; and the 127th is 127x64 = 8128.

In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence every triangular number is either divisible by three or has a remainder of 1 when divided by nine:

0 = 3×0,

1 = 9×0+1,

3 = 3×1,

6 = 3×2,

10 = 9×1+1,

15 = 3×5,

21 = 3×7,

28 = 9×3+1,

36 = 9×4,

45 = 9×5,

55 = 9×6+1,

...

The digital root pattern, repeating every nine terms, is "1 3 6 1 6 3 1 9 9".

The inverse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.

If x is a triangular number, then ax+b is also a triangular number, given the following conditions are satisfied:

a=an odd square, b=(a-1)/8

Note that b will always be a triangular number, because 8T_n+1=(2n+1)2, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for b given a is an odd square is the inverse of this operation.

The first several pairs of this form (not counting 1x+0) are: 9x+1, 25x+3, 49x+6, 81x+10, 121x+15, 169x+21,.... . Given x is equal to T_n, these formulas yield T_3n+1, T_5n+2, T_7n+3, T_9n+4, and so on.

The sum of the reciprocals of all the nonzero triangular numbers is:

This can be shown by using the basic sum of a telescoping series:

Two other interesting formulas regarding triangular numbers are:

and

both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.

In 1796, German mathematician and scientist Carl Friedrich Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers, writing in his diary his famous words, "EΥΡHKA! num = Δ + Δ + Δ" Note that this theorem does not imply that the triangular numbers are different (as in the case of 20=10+10), nor that a solution with three nonzero triangular numbers must exist. This is a special case of Fermat's Polygonal Number Theorem.

The largest triangular number of the form 2k-1 is 4095, see Ramanujan–Nagell equation.

Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression. It was conjectured by Polish mathematician Kazimierz Szymiczek to be impossible. This conjecture was proven by Fang and Chen in 2007.