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3 votes

Irreducible elements in $\mathbb{Z}[\sqrt{5}]$

Throughout this argument, we will use the fact that the norm is multiplicative, that is, $N(x)N(y)=N(xy)$ (I implore you to check this). A unit is an element $x$ with $N(x)=1$. An element $x$ is ...
ljfirth's user avatar
  • 112
3 votes
Accepted

Sum of three squares equalling a different sum of three squares

There is a classical way of finding all rational solutions to a quadratic equation (such as the Pythagorean equation $x^2+y^2=1$). Fix one solution—in our case we'll fix $(x_1,x_2,x_3,y_1,y_2,y_3) = (-...
Greg Martin's user avatar
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2 votes

Combinatorial rectangle packing problem

Suppose that $A(n)=m^2$ for some natural numbers $n$ and $m$. That is $$n(n+1)(n+2)(3n+1)=24m^2.$$ This Diophantine equation can be elementarily split into the system of Generalized Pell's equations ...
Alex Ravsky's user avatar
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2 votes

Diophantine taxi numbers : $n = p q r = a^3 + b^3 = c^3 + d^3 $ for primes $p,q,r$ (example $1729$)

As listed in https://oeis.org/A272935, there are indeed other values of $n$. For example, \begin{array}{rcccccl} 20683 &=& 10^3 + 27^3 &=& 19^3 + 24^3 &=& 37 \cdot 43 \cdot 13, ...
VTand's user avatar
  • 2,689
2 votes
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Is there a method to decide if a diophantine equation has solutions within a given region?

Let us indeed use the simplified equation $(x-y)(x+y)=x^2-y^2=n$ and restrict the domain of search to $\big(\sqrt{n},\tfrac{n}{2}\big)\times\big(0,\tfrac{n}{2}\big)$. Case 1: $n=4$. This clearly has ...
AnCar's user avatar
  • 1,573
2 votes

How to determine the positive odd integers $a$ such that $\forall b\in\{2,4,6,8\}$, $a^6+b$ is not prime?

Every integer $a$ of the form $a=210k+1$ (with $k\ge1$) has this property. Specifically: $a^6$ is obviously not prime; $a\equiv1\pmod3$, so $a^6+2\equiv1^6+2\equiv0\pmod3$, and so $3\mid(a^6+2)$; $a\...
Greg Martin's user avatar
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1 vote
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The Diophantine equation $45x^4-42x^2y^2+5y^4=8$

Here is a practical answer, for the theory or algorithm description check the links, or someone else can elaborate if needed. This is a Thue equation and there are efficient algorithms for finding all ...
Sil's user avatar
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1 vote
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Integer solutions for $2abcd=(a+2)(b+2)(c+2)(d+2)$.

First divide through and rewrite the equation as $$\left(1+\frac{2}{a}\right)\left(1+\frac{2}{b}\right)\left(1+\frac{2}{c}\right)\left(1+\frac{2}{d}\right)=2$$ It's usually a good idea with these ...
Chris Lewis's user avatar
  • 2,248
1 vote

Integer solutions for $2abcd=(a+2)(b+2)(c+2)(d+2)$.

If all $a,b,c,d \geq 11$, then we have that $$2abcd = \prod_{cyc}(a+(\sqrt[4]{2}a-a)) > \prod_{cyc}(a+2) =(a+2)(b+2)(c+2)(d+2)$$
Aaa Lol_dude's user avatar
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1 vote
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Please help me solve this polynomial equation. I would like to know how many integer solution it has?

First note that no solution can be of the form $(-5,y)$ (you should check this). Since your polynomial is linear in $y$, we can solve for $y$ and get $$y=\frac{8x^3-x-13}{x+5}=8x^2-40x+199+\frac{982}{...
ShyamalSayak's user avatar

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