Form 2702: Fill & Download for Free

The answer I found is the triangle with sides [math]2701[/math], [math]2702[/math][math][/math] and [math]2703[/math] units.Let [math]m=n+1[/math], so that the triangle has sides [math]m-1[/math], [math]m[/math] and [math]m+1[/math]. (This makes the calculations slightly easier, in my view.) By Heron’s formula, the area of a triangle of sides [math]m-1[/math], [math]m[/math] and [math]m+1[/math] is[math]\sqrt{s(s-m+1)(s-m)(s-m-1)}[/math]where [math]s=\dfrac{m-1+m+m+1}{2}=\dfrac{3m}{2}[/math]. The area formula thus simplifies to[math]A=\sqrt{\dfrac{3m}{2}\left(\dfrac{m}{2}+1\right)\left(\dfrac{m}{2}\right)\left(\dfrac{m}{2}-1\right)}[/math][math]A=\dfrac{m\sqrt{3}}{2}\sqrt{\dfrac{m^2}{4}-1}.[/math]We require the smallest [math]m[/math] such that [math]A[/math] is a multiple of [math]20[/math]. Of course, we first require [math]A[/math] to be an integer. This happens when the expression [math]\dfrac{m^2}{4}-1[/math] is three times a square, so that [math]z=\sqrt{3}\sqrt{\dfrac{m^2}{4}-1}[/math] is an integer.By making an Excel sheet, I noticed that the integers of [math]x[/math] from [math]1[/math] to [math]209[/math] such that [math]3x^2+1[/math] is a square are [math]1,4,15,56[/math] and [math]209[/math]. Unfortunately, none of these values give me an area that is a multiple of [math]20[/math]. I then noticed that this sequence of numbers seems to have the following recurrence relation:[math]a_1=1[/math][math]a_2=4[/math][math]a_{n+2}=4a_{n+1}-a_n.[/math]By this rule, the next number in the sequence is [math]780[/math]. When [math]x=780, 3x^2+1=1825201[/math], which is indeed a square [math](1351^2)[/math]. Thus [math]m=1351\times 2=2702[/math] is the next number to try. For this number,[math]A=\dfrac{2702\sqrt{3}}{2}\sqrt{\dfrac{2702^2}{4}-1}=3161340[/math]which is a multiple of [math]20[/math]. Thus, [math]m=2702[/math], or [math]n=2701[/math], is the smallest number having this property.The only blank I need to fill is to prove that the recurrence relation I spoke about earlier does indeed give me the next square of the form [math]3x^2+1[/math]. The proof lies in the fact that the integer solutions of the equation [math]y^2-3x^2=1[/math] happen to be a well-studied case of the Pell equation.

Not like a textbook where it explains stuff, but these two books are deep, in my opinion, and very good reads (just fyi, these books take some time to read, but are definitely worth it).1) Gödel, Escher, Bach: An Eternal Golden Braid: Douglas R. Hofstadter: 9780465026562: Amazon.com: BooksShort summary:Twenty years after it topped the bestseller charts, Douglas R. Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid is still something of a marvel. Besides being a profound and entertaining meditation on human thought and creativity, this book looks at the surprising points of contact between the music of Bach, the artwork of Escher, and the mathematics of Gödel. It also looks at the prospects for computers and artificial intelligence (AI) for mimicking human thought. For the general reader and the computer techie alike, this book still sets a standard for thinking about the future of computers and their relation to the way we think.Hofstadter's great achievement in Gödel, Escher, Bach was making abstruse mathematical topics (like undecidability, recursion, and 'strange loops') accessible and remarkably entertaining. Borrowing a page from Lewis Carroll (who might well have been a fan of this book), each chapter presents dialogue between the Tortoise and Achilles, as well as other characters who dramatize concepts discussed later in more detail. Allusions to Bach's music (centering on his Musical Offering) and Escher's continually paradoxical artwork are plentiful here. This more approachable material lets the author delve into serious number theory (concentrating on the ramifications of Gödel's Theorem of Incompleteness) while stopping along the way to ponder the work of a host of other mathematicians, artists, and thinkers.The world has moved on since 1979, of course. The book predicted that computers probably won't ever beat humans in chess, though Deep Blue beat Garry Kasparov in 1997. And the vinyl record, which serves for some of Hofstadter's best analogies, is now left to collectors. Sections on recursion and the graphs of certain functions from physics look tantalizing, like the fractals of recent chaos theory. And AI has moved on, of course, with mixed results. Yet Gödel, Escher, Bach remains a remarkable achievement. Its intellectual range and ability to let us visualize difficult mathematical concepts help make it one of this century's best for anyone who's interested in computers and their potential for real intelligence. --Richard DraganTopics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.2) Cryptonomicon: Neal Stephenson: 9780060512804: Amazon.com: BooksShot summary :In 1942, Lawrence Pritchard Waterhouse—mathematical genius and young Captain in the U.S. Navy—is assigned to detachment 2702. It is an outfit so secret that only a handful of people know it exists, and some of those people have names like Churchill and Roosevelt. The mission of Waterhouse and Detachment 2702—commanded by Marine Raider Bobby Shaftoe-is to keep the Nazis ignorant of the fact that Allied Intelligence has cracked the enemy's fabled Enigma code. It is a game, a cryptographic chess match between Waterhouse and his German counterpart, translated into action by the gung-ho Shaftoe and his forces.Fast-forward to the present, where Waterhouse's crypto-hacker grandson, Randy, is attempting to create a "data haven" in Southeast Asia—a place where encrypted data can be stored and exchanged free of repression and scrutiny. As governments and multinationals attack the endeavor, Randy joins forces with Shaftoe's tough-as-nails granddaughter, Amy, to secretly salvage a sunken Nazi submarine that holds the key to keeping the dream of a data haven afloat. But soon their scheme brings to light a massive conspiracy with its roots in Detachment 2702 linked to an unbreakable Nazi code called Arethusa. And it will represent the path to unimaginable riches and a future of personal and digital liberty...or to universal totalitarianism reborn.

Arguably, there was. The Army recruited artists, designers, and other creatives to form a unit called the 23rd Headquarters Special Troops, later dubbed the Ghost Army. While the motivations for deception apparently had more to do with fomenting confusion than masking the defeat of Enigma encryption, their actions were similar to 2702’s. They used props, sound, and radio to simulate fictitious movements of real units, masking the actual units’ objectives.I learned about the 23rd from an episode of the excellent design podcast 99% Invisible. Link: Show of Force - 99% InvisibleAnd here’s an inflatable tank! The 23rd had to be careful not to be seen performing superhuman feats with inflatables, such as carrying them across open spaces, and they had to use bulldozers to simulate tank tracks for the benefit of German aerial surveillance.