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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.

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3 Answers 3

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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.

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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.

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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.

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  • $\begingroup$ LaTex would make your answer much more readable. $\endgroup$ Commented Apr 21, 2016 at 1:21
  • $\begingroup$ $1$ is not a prime. $\endgroup$ Commented Apr 21, 2016 at 1:53

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