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