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Let us understand the term nearly isosceles Pythagorean triple to mean one whose legs differ by $1.$ A fortiori such a triple is primitive.

After someone asked me how to find such triples, it was easy to derive the proposition labeled below with the word Proposition.

But then in the table below I noticed a pattern that I had not expected:

Pattern:

In the table below, alternate denominators in the first column, $5,29,169,985,$ are the same as the sequence of hypotenuses. Maybe "forever"?

I'm guessing this may be as easy to prove as the "Proposition" below the "Pattern" below, but

QUESTIONS:

  • Reference request: Is this "Pattern" in "the literature"? NOTA BENE: The odd-indexed denominators! (The posted answers by David Richter and KConrad show no awareness that that's what the question is about.)
  • How much reason is there to care about this "Pattern"? Might it have interesting consequences in geometry or number theory or something?

Proposition: Let $x/y$ be a convergent in the simple continued-fraction expansion of $\sqrt2$, in lowest terms. Then $x^2-2y^2\in\{\pm1\}$ and $(x^2+2xy,\,2xy+2y^2,\, x^2+2xy+2y^2)$ is a nearly isosceles Pythagorean triple.

$$ \begin{array}{|c|r|c|ccccccc} \hline & & \text{Nearly isosceles} \vphantom{\dfrac11} \\ x/y & x^2-2y^2\in\{\pm1\} \quad & \text{Pythagorean triple} \\[2pt] \hline \vphantom{\dfrac11} 1/1 & 1^2 - 2\cdot1^2 = -1 & 3^2+4^2=5^2 & \\[8pt] 3/2 & 3^2 - 2\cdot2^2 = +1 & 21^2+20^2=29^2 \\[8pt] 7/5 & 7^2-2\cdot5^2 = -1 & 119^2 + 120^2=169^2 \\[8pt] 17/12 & 17^2 - 2\cdot12^2 = +1 & 697^2 + 696^2 = 985^2 \\[8pt] 41/29 & 41^2 - 2\cdot29^2 = -1 & 4059^2 + 4060^2 = 5741^2 \\[8pt] 99/70 & 99^2 - 2\cdot70^2 = +1 & 23661^2+ 23660^2= 33461^2 \\[8pt] 239/169 & 239^2-2\cdot169^2 = -1 & 137903^2+ 137904^2= 195025^2 \\[8pt] 577/408 & 577^2-2\cdot408^2 = +1 & 803761^2+ 803760^2= 1136689^2 \\[8pt] 1393/985 & 1393^2-2\cdot985^2 = -1 & 4684659^2+ 4684660^2= 6625109^2 \\ \vdots & \vdots\,\,\,\phantom{+1} & \vdots \vphantom{\dfrac{\sum}{\sum}} \\ \hline \end{array} $$

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  • $\begingroup$ I think it is known, I have seen it somewhere, but can't provide you reference. $\endgroup$
    – Somnium
    Jul 7, 2022 at 18:50
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    $\begingroup$ $5=1^2+2^2$; $29=2^2+5^2$; $169=5^2+12^2$; $985=12^2+29^2$; and so on. $\endgroup$ Jul 7, 2022 at 23:39
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    $\begingroup$ @J.W.Tanner : Those are not nearly isosceles Pythagorean triples; rather they are Pythagorean triples in which one leg differs from the hypotenuse by $1. \qquad$ $\endgroup$ Jul 7, 2022 at 23:56
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    $\begingroup$ @J.W.Tanner : Your example gives triangles that are "nearly isosceles" in one plausible common-sense understanding of the term, but not by the definition that I gave. Perhaps that suggests that I should have chosen a different term. $\endgroup$ Jul 7, 2022 at 23:59
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    $\begingroup$ @MichaelHardy: I see what you’re saying about the triples in my now-deleted comment , $\left(k,\dfrac {k^2-1}2, \dfrac{k^2+1}2\right)$, with $k$ a positive odd integer $\endgroup$ Jul 8, 2022 at 0:14

3 Answers 3

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You might read about the relation between Pell's equation and Pythagorean triples. There is some discussion of this at the Mathematics StackExchange, for example. The discussion over there has some references. I don't know much about the geometric or number-theoretic significance of this.

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  • $\begingroup$ Does that discussion include something about the alternate denominators (that being what my question is about)? $\endgroup$ Jul 8, 2022 at 19:06
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$r^2+(r+1)^2=s^2$ is equivalent to $(2r+1)^2-2s^2=-1$ which in turn is equivalent to $\displaystyle{2r+1\over s}$ being an even convergent to $\sqrt2$. That's why the alternate denominators of convergents are the near-isosceles hypotenuses.

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    $\begingroup$ As for references to the literature, I suspect that, being so close to the surface, many would have noticed it but not bothered to draw attention to it. $\endgroup$ Jul 10, 2022 at 3:17
  • $\begingroup$ Albert H Beiler, Recreations in the Theory of Numbers, goes on at some length about nearly isosceles Pythagorean triples. Pages 122 to 125 of the 1964 Dover paperback. He returns to the topic on pages 255 to 257, with regard to $x^2-2y^2=-1$. The very next topic in the book is continued fractions, and that for $\sqrt2$ is exhibited, but the connection isn't made. Amusingly, in the index, the very next entry after "Fractions, continued" is "Fractions–$\it cont$." $\endgroup$ Jul 11, 2022 at 4:35
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    $\begingroup$ I will come back and digest this answer carefully later this week. $\endgroup$ Jul 12, 2022 at 4:56
  • $\begingroup$ How is the digestion process coming along, Michael? $\endgroup$ Jul 21, 2022 at 3:25
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    $\begingroup$ I've been away for a bit. To be continued . . . . . $\qquad$ $\endgroup$ Jul 21, 2022 at 22:09
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See Application 5 (slides 25-27) here, which is part of a talk I have given numerous times on Pell’s equation (most recently yesterday). I think I first learned the application you ask about in Ed Barbeau’s book Pell’s Equation.

There is no real importance to this topic at all. It is just a simple geometry problem that unexpectedly is equivalent to solving a particular negative Pell’s equation, $x^2-2y^2=-1$.

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  • $\begingroup$ Actually it's slides 25–26. But those do not address the question that I asked. Are you aware of the question that I asked? Your posting does not seem to indicate that you are. $\endgroup$ Jul 8, 2022 at 17:43
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    $\begingroup$ Yes, slide 27 was not immediately relevant. Since David Richter and I both missed the point of your post, I suggest editing it to be much clearer what your main question is. It looks like Gerry Myerson explained in a concise way the pattern you observed. I think the only value in such things is pedagogical. You can ask lots of questions about Pythagorean triples and often they are unimportant for modern mathematics except as simple examples to show students in a class (how a mathematician may notice a pattern). $\endgroup$
    – KConrad
    Jul 9, 2022 at 6:09

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