$\begingroup$

I've been trying to figure this out and it's been getting on me myself. I know that $3$ is not just a prime number, but also a triangular number. I'll now add a sequence:

Prime numbers: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107$ Triangular numbers: $1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300$

Anyway, let's cut to the chase. Does this sequence help anything about which prime numbers are also triangular numbers? Now I want to know from you, yeah, you. How many prime numbers can also be triangular numbers? I don't think it's probable. If you have serious, stupendous answers, I would be glad to accept one of them.

$\endgroup$

3 Answers

$\begingroup$

The triangular numbers have the form $\cfrac {n(n+1)}2$. If $n\gt 2$ then whether it is $n$ or $n+1$ which is even, the triangular number has a factorisation into two integers both greater than one, and can't be prime.

$\endgroup$ $\begingroup$

Every triangular number can be written in the form $T_n = (1/2)n(n+1)$, which can be simplified, because either $n$ or $n+1$ is even, so we can remove the factor of $1/2$ and see that $T_n$ can be factorised. This works except for all $n > 2$, hence $3$ is the only prime triangular number.

$\endgroup$ $\begingroup$

The only number that is triangle and prime is $3$. The triangle numbers can be generated from $T= 2n^2\pm n = n(2n\pm 1)$ so that when $n=1$, triangle number $3$ is generated, and when $n=2$, triangle numbers $6$ and $10$ are generated, etc. From this, you can see that triangle numbers can always be factored into the $n(2n\pm1)$ form, notwithstanding $n=1$ as a factor.

Hope this helps.

$\endgroup$ 2

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy