# Questions tagged [pythagorean-triples]

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### Reference request on a pattern among nearly isosceles Pythagorean triples

Let us understand the term nearly isosceles Pythagorean triple to mean one whose legs differ by $1.$ A fortiori such a triple is primitive. After someone asked me how to find such triples, it was easy ...
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### Size of set of integers with all sums of two distinct elements giving squares

Are there arbitrarily large sets $\mathcal S=\{a_1,\ldots,a_n\}$ of strictly positive integers such that all sums $a_i+a_j$ of two distinct elements in $\mathcal S$ are squares? Considering subsets in ...
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### Finding Pythagorean quadruples on a given plane?

In 2D one cannot construct Pythagorean triples $x^2+y^2=m^2$ ($x,y,m\in\mathbb{Z}$) that lie on every line through the origin (e.g., a Pythagorean triple with $x=y$ would require $\sqrt{2}$ to be ...
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### Triangulating the plane using edges of unique rational lengths

Basic question: Can the Euclidean plane be divided into a vertex-to-vertex arrangement of non-overlapping triangles such that every edge has a unique rational length that lies between 1 and some ...
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### Prove $\frac{\text{Area}_1}{c_1^2}+\frac{\text{Area}_2}{c_2^2}\neq \frac{\text{Area}_3}{c_3^2}$ for all primitive Pythagorean triples

A while ago I asked this question on MSE here. After placing a bounty it got quite a bit of attention but unfortunately it has yet to be resolved. After getting some advice from MO Meta I have decided ...
Numbers of the form $4mn-m-n$ where $m,n\in\mathbb{Z}^+$ are $$A=\{2, 5, 8, 11, 12, 14, 17, 19, 20, 23, 26, 29, 30, 32, 33, 35, \ldots\}$$ The set complement of the above set is $$B=\{1, 3, 4, 6, ... 7 votes 0 answers 303 views ### Is 100 the only Leyland number that is a square? Leyland numbers (named for Paul Leyland) are positive integers of the form x^y + y^x , where x and y are naturals > 1, and also the number 3. The OEIS link is https://oeis.org/A076980 I thought ... 8 votes 1 answer 338 views ### Integer points avoiding three on a line, four on a circle A century ago, Dudeney asked to place 16 pawns on a chessboard with no three on a line: As described by David Eppstein,1 the maximum number g_3(n) points that ... 5 votes 2 answers 406 views ### Can we surround a non-rectangular area with Lego fences? My children have some Duplo fences, these you have to put down on two points, and at both ends they extend a little where you can connect several to surround some area. So a fence is described by a ... 9 votes 2 answers 2k views ### Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2? Given j \geq 5, is there a formula for the number of Pythagorean triplets (a, b, c) satisfying the constraint that a, b, c \leq j? There exists at least one Pythagorean triplet for j\geq5; ... 13 votes 1 answer 582 views ### Realization of numbers as a sum of three squares via right-angled tetrahedra De Gua's theorem is a 3-dimensional analog of the Pythagorean theorem: The square of the area of the diagonal face of a right-angled tetrahedron is the sum of the squares of the areas of the other ... 2 votes 0 answers 101 views ### Primitive triples in a region [duplicate] Are there at least cn Pythagorean triples and at least dn Primitive Pythagorean triples (A,B,C) with$$2^{\frac n2}<A<2^{\frac n2+1}<2^n<B<C<2^{n+1}$$with some fixed 0<d&... 1 vote 2 answers 214 views ### Sharply Estimating Pythagorean Triples [closed] Given m,n\in\Bbb N with m<n, how many pythagorean triples p^2+r^2=q^2 satisfy$$m\leq p<r\leq n? Is there a way to give a sharp estimate? 