25
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 ...

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 ...

18
votes

Accepted

### Can the Pythagorean Graph be finitely colored?

This paper shows that the chromatic number is infinite.
Indeed, Theorem 1.1 part (i) with $a=b=c=1$ is what you want.

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+...

8
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 ...

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 ...

5
votes

### Pythagorean triples and quadratic residues modulo primes

The Conjecture 1 is true. We are looking for integers $m, n$ such that for sufficiently large prime $p$ we have
$$
x_{1}^2\equiv 2mn\pmod{p},\quad x_{2}^2\equiv m^2-n^2\pmod{p}, \quad x_{1}^2\equiv m^...

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 ...

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 ...

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 ...

3
votes

### 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 ...

2
votes

### Is 100 the only Leyland number that is a square?

Here are two unconditional results:
if $x$ is a power of $2$ then $(x,y)=(2,6)$
$\gcd(x,y)=1$ or $2$.
I thought of this problem in 2019 and posted it on MSE (link), as at that time I did not ...

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 ...

2
votes

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

It is an old question but I recently solved a programming problem regarding this question, the article was shared above but the method was not explicitly shared here.
Following this article: ...

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 ...

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 ...

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_{...

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 ...

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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