In the sequence of triangular numbers, some numbers are twice another.

For example t(20)=210 which is twice t(14)=105.

Characterize all such numbers.

Easy bonus: Explain why (except for the trivial case) there are no square numbers that are twice another.

Lemma:

The equation

** x^2=2y^2+1 ** (Pell eq.)

has an infinite number of integer solutions.

Proof:

Clearly **(1,0)** is a solution.

It is obvious that if **(x,y)** is a solution so is **(3x+4y,** ** 2x+3y),** therefore ** ****(1,0)** generates a chain of valid solutions, each derived from the previous one:

** (1,0); (3,2); (17,12); (99,70);… etc**

Back to our problem:

**n/m=2*(m+1)/(n+1)** let us call **n/m=p/q**

so ** n=pm/q** and **n+1=2q*(m+1/p)**

and **m(p^2-2q^2) = 2q^2-pq**

Evaluating **(p,q)** by solving the **Pel. Eq.**

we get integer values for m and n:

**m=q*(2q-p)** and **n= p*(2q-p) **

Example:

** (p,q)=(17,12**) generates **( m,n)=(84,119**)

Bonus question, having nothing in common with the topic discussed above, can be resolved by an easy proof:

If m and n are integers (any inegers) then m/n is a ratoinal number, but **m^2/n^2=2** forces

** m=n*sqrt(2)** which is irrational.

** Proven by contradiction.**

*Edited on ***February 19, 2015, 3:31 pm**