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August 9

Pythagorean triple angles

Is it coincidence that the larger acute angle of a 7-24-25 Pythagorean-triple triangle is twice the smaller acute angle of a 3-4-5 triangle? Thanks, cmɢʟee τaʟκ (please add {{ping|cmglee}} to your reply) 00:58, 9 August 2026 (UTC)

No, it is not a coincidence. See trigonometric formulas#Double-angle formulae.

$\tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}$.

and apply that. JRSpriggs (talk) 01:53, 9 August 2026 (UTC)

In fact, if (a, b, c) and (d, e, f) are Pythagorean triples, then so is (ad-be, ae+bd, cf) (provided ad-be>0). The angle opposite the ae+bd side will then be the sum of the angles opposite the b and e sides in the original triangles. The 7-24-25 is a special case, the "product" of the 4-3-5 triangle with itself. --RDBury (talk) 08:39, 9 August 2026 (UTC)

I naively expected that iterating $(a,b,c)\mapsto (|a^{2}-b^{2}|,2ab,c^{2}),$ starting with $(3,4,5),$ would soon converge to some set of angles. Instead, this keeps jumping all over the place. Then I realized this corresponds to iterating $t\mapsto |2t^{2}-1|,$ where $t=a/c.$ This map has the sole fixpoint ${\tfrac {1}{2}}$ in $(0,1),$ which is not stable, making convergence impossible.  ​‑‑Lambiam 10:30, 9 August 2026 (UTC)

Thanks very much, everyone. cmɢʟee τaʟκ (please add {{ping|cmglee}} to your reply) 11:40, 9 August 2026 (UTC)

Group of rational points on the unit circleRich (talk) 18:41, 9 August 2026 (UTC)

August 13

What does "a priori" mean in a mathematical context?

I am struggling with how to rephrase "a priori" in the draft I created.--SilverMatsu (talk) 02:35, 13 August 2026 (UTC)

In analysis, one usually uses the term in reference to what can be inferred from a problem without knowing its particular solution. The basic instance is an a priori estimate. For example, if you are studying an equation like $Lu=f$ where $L$ is an elliptic differential operator, you can prove an inequality like $\|u\|\leq C\|f\|$ (with suitable norms on both sides) without knowing anything else about a solution (or even whether it exists). Sławomir Biały (talk) 07:31, 13 August 2026 (UTC)

That the range of f is the domain of g does not guarantee that the composition g°f exists. JRSpriggs (talk) 13:43, 13 August 2026 (UTC)

Paper

• Coppey, L. (1980). "Quelques problèmes typiques concernant les graphes multiplicatifs" (PDF). Diagrammes (in French). 3 (2). ISSN 0224-3911. -The phrase "a priori" appears at the introduction of this paper.