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I'm a beginner in math (but, know some high school math concepts) and I read the book A Friendly Introduction to Number Theory for the basis of cryptography, but something confused me in section 1 in the Numbers Shape section, the author makes a question: "A natural question to ask is whether there are any triangular numbers that are also square numbers (other than 1)." what makes me confused is, the author tries to share the formula from the Gauss formula, which if I calculate the result only produces triangular numbers and may not get a value for square numbers. the author also makes illustration but makes me even more confused. The author makes a square shape, although when \begin{align}(n + 1)\end{align} is placed between \begin{align}\frac{n(n + 1)}{2}\end{align} it will be seen that the square is also the result of 2 triangles, but is that what is meant by a number that can not only be triangular but also square? thank you

syafiqfadillah
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  • A triangular number is as you say n(n+1)/2. A square number is just m^2 for some m. So if a triangular number is also a square number, then the equation n(n+1)/2=m^2 holds. This is certainly so if n=1 and the "natural question" is asking if there are more solutions. – coffeemath Apr 18 '24 at 10:26
  • Not sure what the question is. Are you asking how that closed formula for the $n^{th}$ triangular number is derived? Are you asking how to find natural numbers which are both triangular and square? Something else? – lulu Apr 18 '24 at 10:26
  • here may be a useful duplicate. – lulu Apr 18 '24 at 10:27
  • $1+2+3+4+5+6+7+8=36=6^2$, so the number $36$ is triangular and is also square. Understood? – Gerry Myerson Apr 18 '24 at 12:33
  • @lulu im sorry if the question was not clear, my question is, can the gauss formula (which is n(n+1)/2) actually generate a number that can be both triangular and square numbers? because as the illustration i gave the author use the gauss formula to prove it, but when i tried n = 7, it was not gave me a proper number. – syafiqfadillah Apr 18 '24 at 13:09
  • Well, $1$ is both triangular and square and a little trial and error should have shown you that so is $36$. A001110 gives the list of them. – lulu Apr 18 '24 at 13:12
  • If one of the numbers $n,n+1$ is a square, and the other is twice a square, then the triangular number $n(n+1)/2$ will be a square. So: $n=49$ is a square, and $n+1=50$ is twice a square, so $49(49+1)/2=49\times25=35^2$. – Gerry Myerson Apr 19 '24 at 10:57
  • Understood, sya? – Gerry Myerson Apr 21 '24 at 05:45
  • @Gerry Myerson I understand the explanation you gave but, I still don't understand what the author of this book wants to convey. currently I am trying another book and I think this book is more understandable than the previous book. and thank you for the explanation and for checking me out! – syafiqfadillah Apr 21 '24 at 10:01
  • You write, "the Gauss formula, which if I calculate the result only produces triangular numbers and may not get a value for square numbers", but I have shown you that the formula that produces triangular numbers can produce squares. You write, "is that what is meant by a number that can not only be triangular but also square?" I've shown you what's meant by a number that is both triangular and a square. I can't read the author's mind, but I hope I have straightened you out on square numbers and triangular numbers. – Gerry Myerson Apr 21 '24 at 12:32
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    @Gerry Myerson thank you! now i've a different views to squre-triangular. lastly, it maybe out of topic, im currently self-studying math for cryptography, i don't have any degree, never had good grades at math and feels like anything from the textbooks like notations, numbers, and explanations are too intimidating. do you have any advice? – syafiqfadillah Apr 22 '24 at 09:11
  • Maybe you could ask for advice at https://matheducators.stackexchange.com/ (but look around first to get an idea of whether the question would be welcome there – I'm not familiar with the site). – Gerry Myerson Apr 22 '24 at 11:56
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    @Gery Myerson i'll check the site, thank you! – syafiqfadillah Apr 22 '24 at 12:17

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