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Does anyone know who was the first to find solutions in integers for $n^2 + (n+1)^2 = k^2$ (almost isosceles Pythagorean triples)? Who proved (if that's the case) that there are no other solutions apart from $$ t = \frac{(3+2\sqrt{2})^n}{4}+\frac{(3-2\sqrt{2})^n}{4}-\frac{1}{2}, \hspace{5mm} d = \frac{(3+2\sqrt{2})^n}{4\sqrt{2}}-\frac{(3-2\sqrt{2})^n}{4\sqrt{2}}?$$

suckling pig
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Vika
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    Are you talking about this? Or this? "Triangular Numbers" have a standard meaning.... – lulu Sep 24 '24 at 20:21
  • Yes, but the questions are 1) who was the first, 2) who proved there are no other solutions? Euler found solutions, there are infinitely many of them, but is it proven that there are no others? – Vika Sep 24 '24 at 20:33
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    Which one? I'm guessing you mean the second one (right triangles) which has nothing to do with "Triangular Numbers". I suggest changing the header of your question. In any case, This instance of Pell's equation has been studied for millennia, as discussed in that link. I don't know that things were properly formalized before Euler and Legrange, but this instance was well understood long before then. – lulu Sep 24 '24 at 20:37
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    @Vika: The dedicated History of Science and Math SE may be a better place for a question like this. ... Cheers! – Blue Sep 24 '24 at 20:49
  • @Lulu: I wasn't able to find the answer if it was proven (and by whom) that there are no other solutions. – Vika Sep 24 '24 at 21:02
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    "No other solutions" other than what? – MJD Sep 24 '24 at 21:05
  • You haven't provided any set of solutions at all. Look up the theory of Pell's equation. Certainly the modern version (continued fractions) is enough. I am not sure whether earlier people working on this had what we'd agree was a rigorous proof or not...I'd be inclined to doubt it, but I also doubt that solid records go back that far. My sense is that they were primarily interested in generating good rational approximations to $\sqrt 2$ and less interested in rigorously proving that they hadn't missed any. But I could have that wrong. – lulu Sep 24 '24 at 21:07
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    It was known during the Old Babylonian period (c. 1800 BCE) that there is a rectangle with sides 119 and 120 and diagonal 169 and a rectangle with sides 3 and 4 and diagonal 5. See https://link.springer.com/article/10.1007/s00407-011-0083-4. – Will Orrick Sep 24 '24 at 22:31
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    Diophantus knew the recipe $x = 2mn, ; y = m^2 - n^2, ; z = m^2 + n^2 ; . ; ; $ The other direction, that every primitive right triangle with integer sides comes from that recipe, is attributed to an anonymous Arabic manuscript of 972 a.d. Using this, all solutions have $m+n$ odd and either $m^2 - 2mn - n^2 = 1$ or $m^2 - 2mn - n^2 = -1.$ That is, $(m+n)^2 - 2 n^2 = \pm 1,$ a Pell equation, as advertised – Will Jagy Sep 25 '24 at 01:34
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    from Dickson, History of the Theory of Numbers, volume II: Diophantine Analysis. Chapter 4, especially pages 165 and 166 – Will Jagy Sep 25 '24 at 02:08
  • @Will Jagy: It is clear for general Pythagorean triples. I need it for "almost isosceles Pythagorean triples" as written in the equation. – Vika Sep 25 '24 at 20:26
  • @MGD: other than the ones provided by Euler, for example (edited the question). – Vika Sep 25 '24 at 20:27
  • The first "almost isosceles Pythagorean triple" is $(3,4,5)$, next is $(20,21,29)$. The formulas presented as solutions for $t$ and $d$ yield $(1,1)$ for $n=1$ and $(8,6)$ for $n=2$ -- alas, I can't see (yet) a relation of $t$ and $d$ with "almost isosceles Pythagorean triples." Please explain for dummies. – m-stgt Sep 26 '24 at 05:12
  • I can show you 3 ways to generate these but the history of where it started is harder to pin down. – poetasis Sep 26 '24 at 23:07
  • While the exact discovery of "nearly isosceles Pythagorean triangles" isn't attributed to a single person, the concept is generally understood within the framework of the Pythagorean Theorem, which is credited to the ancient Greek mathematician Pythagoras; however, evidence suggests that knowledge of Pythagorean triples (including those that could be considered "nearly isosceles") existed in ancient Babylonian mathematics before Pythagoras' time. – poetasis Sep 27 '24 at 21:40

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