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 😀

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