23 votes
Accepted

Size of set of integers with all sums of two distinct elements giving squares

The size of such sets is bounded by some (unknown) constant, assuming a big conjecture in arithmetic geometry. The Bombieri-Lang conjecture (non-trivially via the Uniformity Conjecture, see Stanley ...
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  • 5,803
19 votes

Size of set of integers with all sums of two distinct elements giving squares

"D15 Numbers whose sums in pairs make squares" in Guy, Unsolved Problems in Number Theory, 3rd ed., credits Erdos and Leo Moser with asking "are there, for every $n$, $n$ distinct ...
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17 votes
Accepted

Realization of numbers as a sum of three squares via right-angled tetrahedra

These numbers are not very prevalent and the ratio in question goes to zero. Note first that by Legendre's theorem, a positive proportion of the numbers below $n$ may be expressed as a sum of three ...
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  • 42.5k
11 votes

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

So you are looking for solutions $$ \bigl( [a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]\bigr) \in (\mathbb P^2)^3(\mathbb Q) $$ to the equations $$ a_1^2+b_1^2=c_1^2,\quad a_2^2+b_2^2=c_2^2,\quad a_3^2+...
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7 votes

Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?

The question here was studied (in a slightly more generalized version) by Sierpinski (1906) in Polish; asymptotics were found by Fricker (1977, 1982) and Fischer (1979) both in German; and an ...
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7 votes
Accepted

Small linear relations between primitive Pythagorean triples $\mathsf{II}$

Yes, the minimal $\|(u,v,z)\|_\infty$ is within a constant factor of $\sqrt{|c|}$ (equivalently, of $\sqrt{\max(|a|,|b|)}$. The orthogonal complement of $(a,b,c) = (m^2-n^2, 2mn, m^2+n^2)$ contains ...
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4 votes
Accepted

Small linear relations between primitive Pythagorean triples $\mathsf I$

Yes, when $m>n>0$ and $$ a = m^2 - n^2 $$ $$ b = 2mn $$ $$ c = m^2 + n^2 $$ then $$ -n a^2 +(m-n)b^2 - n ab +(m-n)bc - n ca = 0 $$ or quintuple $$ -n, m-n, -n, m-n, -n $$ There is a second ...
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  • 24.5k
4 votes

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

Not really an answer, but a suggestion. Did you try to solve the "wrong" problem, where at the denominators you have $c_i$, not $c_i^2$? It seems quite interesting. If we restrict ourselves to ...
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3 votes

Size of set of integers with all sums of two distinct elements giving squares

This is more of a comment than an answer. This problem has been considered by many, but the problem is that the solution is reduced to solving a system of Diophantine equations of the 2nd degree. Even ...
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  • 396
2 votes

Integer points avoiding three on a line, four on a circle

this should be considered as a comment: There seems to be a strong relation of the "no 4 points cocircular" problem to pythagorean triples: Taking $(0,0)$ as one of the corners of Pythagorean ...
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  • 11.2k
1 vote

Reference request on a pattern among nearly isosceles Pythagorean triples

$r^2+(r+1)^2=s^2$ is equivalent to $(2r+1)^2-2s^2=-1$ which in turn is equivalent to $\displaystyle{2r+1\over s}$ being an even convergent to $\sqrt2$. That's why the alternate denominators of ...
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1 vote

Reference request on a pattern among nearly isosceles Pythagorean triples

You might read about the relation between Pell's equation and Pythagorean triples. There is some discussion of this at the Mathematics StackExchange, for example. The discussion over there has some ...
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1 vote

Finding Pythagorean quadruples on a given plane?

It is not the answer but some relevant information. In the paper "Cubes in an Integer Lattice" Ivan Horozov gave parameriyation of all mutually perpendicular integer vectors $A_{1}=\left(x_{...
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1 vote

Triangulating the plane using edges of unique rational lengths

We can at least triangulate an infinite strip with bounded rational side lengths that do not repeat. We take the strip between $x=0$ and $x=1$, alternating points on each line. The distances will all ...
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  • 17.1k
1 vote

Сomplement of the set of numbers of the form $ 4mn - m - n$?

Here are my thoughts. Let us characterize $A=\{4mn-m-n:m,n\in\mathbb{Z}^+\}$ more precisely. Note that, $k\in A \Leftrightarrow 4k+1=(4m-1)(4n-1)$ for some positive integers $m,n$. Since $m,n$ are ...
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  • 1,076
1 vote

Can we surround a non-rectangular area with Lego fences?

If I understand the geometry of the fences correctly, it should be possible to surrounded a triangle of dimension 15 by 20 by 25, each of which is a multiple of 5. In general, if 5 is replaced with $n$...
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1 vote

Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?

The problem is equivalent to asking for all Pythagorean triples with bounds on the hypotenuse and a variant of that problem is treated in the article ENUMERATION OF ALL PRIMITIVE PYTHAGOREAN TRIPLES ...
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  • 11.2k
1 vote
Accepted

Sharply Estimating Pythagorean Triples

See my preprint. Page 12 (this has since been published, but preprint is easiest to access).
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