# Triangles and Squares

Here’s a quick one for you. Consider the sequence of triangular numbers

$1, 3, 6, 10, 15, 21,28,36,45, \ldots$

and the set of square numbers

$1, 4, 9, 16, 25, 36, 49,\ldots$

It seems that there are some numbers which are both squares and triangles! For example, 1 and 36 get to be both shapes – we will call these square-triangles from now on.

Actually, you can very easily prove that there are infinitely many square-triangles via the following exercise.

Exercise. Let $m$ be a square-triangle number. Prove that $4m$ and $8m+1$ are both square numbers. Finally, prove that $4m (8m+1)$ is a square-triangle.

## 4 thoughts on “Triangles and Squares”

1. This proof uses the same kind of method which we use to show infinite primes. Considering p1 and p2 to be primes, p1*p2 +1 will always be a prime p3. Now p1*p2*p3 +1 is the next prime p4. Does this method of proving have a name?
Nice article!

2. Thanks for the comment, but you are not quite correct. If I choose, for example, p1 = 3 and p2=5, then p1*p2+1 = 16 which is not a prime. The way we prove that there are infinitely many primes (at least, traditionally) is as follows:

Assume that there are only finitely many prime numbers which we can list as p1, p2, … , pk. If we let N=p1 x p2 x … x pk, then it is clear that N+1 must be divisible by a prime that we haven’t listed, for any two consecutive integers can have no prime factors in common.

I don’t actually think there are any “simple” functions which take in a set of primes and give you a new prime, though I am happy to be corrected!

1. Thanks for correcting me! I should have thought about that before commenting. That new number is not divisible by any primes we had considered before. But we don’t have a quantitative description of what that prime number is. I realize this now!
And by the way, I would love to hear something about primes from you. That has always fascinated me! 😀

1. No worries! I think I wrote a couple of posts on prime numbers a year or so ago. I will have to write something new. Thanks for the good feedback 😀