How Paul Erdős Cracked This Geometry Problem | The Anning-Erdős Theorem

Are there infinitely many points, not all on the same line, that are an integer distance apart? The answer is given by the Anning-Erdős theorem. In this video, we prove their theorem. To stay up to date, consider subscribing to this YouTube channel. Thanks to my daughter, Meena Boppana, for reviewing this video. She is a math tutor: http://meenaboppana.com Chapters 00:00 Introduction 01:26 100 Points 07:02 Infinitely Many Points 07:54 The Anning-Erdős Theorem 09:53 Proof of the Anning-Erdős Theorem 15:18 Intersection Points of Conic Sections Wikipedia article on the Anning-Erdős theorem: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Anning_theorem Norman H. Anning and Paul Erdős (1945), "Integral distances", Bulletin of the American Mathematical Society 51(8), pages 598–600. https://www.ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/S0002-9904-1945-08407-9.pdf Paul Erdős (1945), "Integral distances", Bulletin of the American Mathematical Society 51(12), page 996. https://www.ams.org/journals/bull/1945-51-12/S0002-9904-1945-08490-0/S0002-9904-1945-08490-0.pdf Wikipedia article on Apollonius of Perga: https://en.wikipedia.org/wiki/Apollonius_of_Perga Thomas Heath's translation/rewrite (1896) of Apollonius's "Treatise on Conic Sections": https://archive.org/details/treatiseonconics00apolrich The result that two distinct conic sections intersect in at most 4 points appears as Proposition 78 on page 130. Wikipedia article on Bézout's theorem: https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem Proof of Bézout's bound in the plane: https://ocw.mit.edu/courses/18-s997-the-polynomial-method-fall-2012/resources/mit18_s997f12_lec13/ We used two hyperbola diagrams by Merrill Sherman from this article in Quanta Magazine on integer distances: https://www.quantamagazine.org/merging-fields-mathematicians-go-the-distance-on-old-problem-20240401/ The final illustration of Paul Erdős is from this book: The Boy Who Loved Math: The Improbable Life of Paul Erdős, written by Deborah Heiligman, illustrated by LeUyen Pham. 2013. Roaring Brook Press. https://deborahheiligman.com/books/the-boy-who-loved-math/ Wikipedia article on the Babylonian clay tablet (Plimpton 322): https://en.wikipedia.org/wiki/Plimpton_322 We displayed part of one table from this article when discussing the 65-72-97 right triangle. The photo of the Babylonian clay tablet came from this Columbia University article: https://magazine.columbia.edu/article/babylon-revisited Thumbnail by Rostislav Demchuk. Several viewers asked this question: how many points can you have, all an integer distance apart, of which no three points are on the same line? The answer: you can have any finite number of such points, all on one circle. Consider the (infinitely many) points on the unit circle with rational coordinates. Viewing them as complex numbers, square them all. The resulting points are a rational distance apart. We can then apply the rational-to-integer trick as in the video. This construction is an "inversion" around a circle of the stack of right triangles from the video.