arXiv · 0804.0152
This paper derives Touchard's theorem from Euler's form for odd perfect numbers. It also fine-tunes Euler's form.
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Eyob Delele Yirdaw. 2008-04-01. Proving Touchard's Theorem From Euler's Form. https://arxiv.org/abs/0804.0152
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In the first part of this work we showed that, within a Hilbert plane deprived of the Side-Angle-Side axiom, the Side-Angle-Angle criterion, together with a ray correspondence principle [\textbf{RCT}], the existence of angle bisectors [\textbf{AB}], the congruence of supplements of congruent angles [\textbf{SA}], and the Pons Asinorum [\textbf{PA}], suffices to reconstruct SAS. We left open the question of whether [\textbf{PA}] is genuinely required alongside the other three principles, noting only a qualitative asymmetry in the nature of the principles involved. In this second part we answer this question: we show that \begin{equation*} \textrm{SAA},\ [\textbf{RCT}],\ [\textbf{AB}] \;\vdash\; [\textbf{PA}], \end{equation*} so that [\textbf{PA}] is redundant among the hypotheses of our main theorem, which improves to \begin{equation*} \textrm{SAA},\ [\textbf{RCT}],\ [\textbf{AB}],\ [\textbf{SA}] \;\vdash\; \textrm{SAS}. \end{equation*} The proof adapts an argument recently given by Donnelly, who reconstructs SAS from SAA together with an angle addition axiom and the existence of angle bisectors.
In {\tt 1878}, Pafnuty Chebyshev presented to the {\it Association fran\c caise pour l'avan\-cement des sciences} {\it [French Association for the Advancement of the Sciences]} an article \cite{Chebyshev1878} dealing with garment cutting. According to Chebyshev himself, his interest was sparked by a lecture given by Édouard Lucas that he had attended in {\tt 1876} \cite{Lucas1876}. There is a second story on the origin of Chebyshev's interest in garment cutting according to which in the 1850s, being short of money, Chebyshev got himself a job as a consultant to a clothing factory. At the time of the Crimean War (1853-1856), there was a great demand for uniforms. Chebyshev was allegedly asked to optimize the use of fabric, and it was there that his interest in garment cutting was born. This second story appears to have its origin in a post by Clive J. Grant to MacTutor in 1996 \cite{Grant1996}. However, this contribution contains no references, and no other source of information that I have found offers any first-hand documentation to support this story. Our conclusion is that this second story is a fabrication, invented out of whole cloth.
Generative AI changes the conditions under which graduate mathematics is learned, assessed, written, and defended. The central claim of this paper is that mathematics graduate programs should respond to the moment by clarifying what graduate mathematics education is trying to teach and assess. In most ways, the goals of mathematics education have not changed. Rather, with changing tools it has become more essential than ever to make clear the goals of mathematical training. We use the term mathematical judgment to refer to the capacity to evaluate mathematics (e.g., claims, definitions, examples, proofs, analogies, computations, uses of tools, research directions) as mathematically sound, useful, well-posed, and appropriately justified. The recommendation is to center training on mathematical judgment, and we examine possible policies for graduate programs in Mathematics to this end.