منابع مشابه
On Two of John Leech’s Unsolved Problems concerning Rational Cuboids
Let {X, Y,Z,A,B, C} ∈ Q+ be such that X2 + Y 2 = A2, X2 + Z2 = B2 and Y 2 + Z2 = C2. We consider the problem of finding T ∈ Q+ such that either 1. T 2 −X2 = , T 2 − Y 2 = , T 2 − Z2 = or 2. T 2 −A2 = , T 2 − B2 = , T 2 − C2 = . We show that problem 2 always has a solution and we provide a formula for T . Extensive computation has been unable to find a single solution of problem 1.
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We consider the set of triangles in the plane with rational sides and a given area A. We show there are infinitely many such triangles for each possible area A. We also show that infinitely many such triangles may be constructed from a given one, all sharing a side of the original triangle, unless the original is equilateral. There are three families of triangles (including the isosceles ones) ...
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In his paper Triangles with three rational medians, about the characterization of all rational-sided triangles with three rational medians, Buchholz proves that each such triangle corresponds to a point on a oneparameter family of elliptic curves whose rank is at least 2. We prove that in fact the exact rank of the family in Buchholz paper is 3. We also exhibit a subfamily whose rank is at leas...
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We show that for all integers m > 4 there exists a 2m × 2m ×m latin cuboid that cannot be completed to a 2m × 2m × 2m latin cube. We also show that for all even m / ∈ {2, 6} there exists a (2m−1) × (2m−1) × (m−1) latin cuboid that cannot be extended to any (2m−1)× (2m−1)×m latin cuboid.
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We prove that there are 14 distinct ways to tile the unit square (modulo the symmetries of the square) with 5 triangles such that the 5-tiling is not a subdivision of a tiling using fewer triangles. We then demonstrate how to construct infinitely many rational tilings in each of the 14 configurations. This stands in contrast to a long standing inability to find rational 4-tilings of the unit sq...
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ژورنال
عنوان ژورنال: Lietuvos matematikos rinkinys
سال: 2018
ISSN: 2335-898X,0132-2818
DOI: 10.15388/lmr.b.2018.9