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In the beginning of the article a
Diophantine equation is defined to be a _polynomial_ equation (with certain properties and conditions), but one of the given examples, namely 4/n = 1/x + 1/y + 1/z, is in fact not a polynomial equation by the definition given in the article about "polynomial" because
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I have used the non-notability for removing your edit because it was the simplest argument. In fact the reason of my revert is more complex. Firstly, I agree that the section "One equation" should be renamed "one equation in two unknowns", and that a section "One equation in more unknowns" could be
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I'm personally not clear on the distinction between
Arithmetic Geometry and Algebraic Geometry, except some vague awareness that the former usually deals with rational points on varieties and the latter with more general algebraic structure. So I can kind of see why 'Arithmetic geometry' redirects
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Fell free to add the other (1980) paper if you think people should read it :-) The only more recent paper on the topic cited in
Schrijver's book is a 1981 paper by Kerztner in AMM. However, I would not suggest adding Kerztner's paper because it turned out unoriginal, i.e. the same as Blankinship's
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For illustrative purposes, we will continuously use the following example with n = 3: (al, a2, a3) = (8913, 5677, 4378). Or, we are interested in the generator of: 8913x1 + 5677x2 + 4378x3 = 0. It turns out that the Bond
Algorithm produces the two generating vectors (5677, -8913, 0) and (2219646,
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To illustrate the discussion, consider a small example with n=3 and a=(8913, 5677, 4378). The algorithm with Rule A produces generating vectors (0, 4378, -5677) and (1, 1, 12736677, -16515789), whereas the procedure with Rule B gives (8189939, -227499, -16378578) and (989097, -27475, -1978037). In
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to this page, since finding rational points on varieties is roughly speaking the same problem as solving
Diophantine equations. But it'd be nice if someone who knows something about the subject could insert a paragraph or two on this page about it, if not create an actual page for it.
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added. However, Knowledge is an encyclopedia, and if a subject is treated, its coverage must not be reduced to a single 35 years old paper, ignoring the more recent results on the subject. Therefore including this article in "See also" section gives a
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The example is trivially modifiable to be a polynomial: 4xyz = yzn + xzn + xyn. So it's not really a violation of the definition, but maybe some explanation of why it isn't might be useful. —
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it contains division. Please someone fix that. Either change the definiton of an
Diophantine equation or remove the offending example. (I dont know which is the correct way)
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Get to work then and add these algorithms (you are so sure exist) to the article on Smith normal form; all I see there is a paper from 1861 (I didn't get the century wrong.)
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in the examples of diophantine equations. I believe it's is a good example because it is both common and easy to solve—using SFFT ("Simon's
Favorite Factoring Trick"):
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If you actually look at the submission date of the "1980" paper it was submitted on May 16, 1978. The 1979 Fibonacci
Quarterly paper is also cited in Schrijver's book:
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Isn't this saying the same thing as "...that restricts two or more variable to integer values only", and if so, isn't the second wording clearer?
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Forty years ago I discovered that the
Fibonacci Sequence (1, 1, 2, 3, 5, 8, etc) can be generated from the second degree Diophantine equation
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tell me in the way to figure out the problem on his tomb theat tells how old he was when he died... please tell me how to do that problem
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The general case of linear
Diophantine equations (any number of unknowns, any number of equations) is treated two subsections later.
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The fact that neither paper cites the other is a bit amusing, but which is the notable one is clear from both their text.
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https://math.stackexchange.com/questions/20906/how-to-find-an-integer-solution-for-general-diophantine-equation-ax-by-cz
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In my opinion, this article deserve a section "One equation in more unknowns", but it must not be based on a single
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I think the article should mention how to expand the solution from two variables to more. Here is a helpful link:
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general, the application of Rule B does not reduce the size of the elements in the generating vectors.
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3484888, -1), whereas the procedure we propose gives (cf. Section 3) (-57, 17, 94) and (61, -95, -1)
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even if the variables were all simply one, wouldn't this be infinite? unless zeros are allowed. --
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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be included in the 'see also' section? somewhat popular among the numberphile crowd. --
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Does someone more qualified want to discuss these? I'm only a high school student.
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The results on complexity and practical efficiency for the computation of the
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The former produces smaller solutions. The "1980" (actually 1978) paper says:
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to this particular paper. Since 1980, the main progresses that I know of are
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algorithm; a note to that effect was published in AMM in 1983
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Fibonacci Sequence and its Second Degree Diophantine Equation
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5k^2 -/+ 4 = m^2 where the -,+ is taken alternately.
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