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Results tagged with diophantine-equations
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user 38851
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
0
votes
6
answers
595
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If $n=x^k+y^k$ then also $n=a^2+b^2=c^3+d^3=\ldots =x^k+y^k$ [closed]
Are there infinitely many positive integers with the property:
If $n$ is a sum of two $k$th powers then it is also the sum of two $k-1$th powers, the sum of two $k-2$th powers, ... , the sum of two sq …
- 3,866
11
votes
3
answers
868
views
How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
It is well known that $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and this is the only solution to $\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}=1$
with $2\leq x_1<x_2<x_3$.
My question is:
Let $n\in …
- 3,866
10
votes
0
answers
221
views
Product of four consecutive primes plus $1$ equals square
Some days ago, I noticed that $3\cdot 5\cdot 7\cdot 11 +1=34^2$.
I am almost sure that if we denote four consecutive primes by $p, q, r, s$ then the equation
$$p\cdot q\cdot r\cdot s+1=x^2 \qua …
- 3,866
16
votes
Accepted
Is there an algorithm to solve quadratic Diophantine equations?
Sierpsinski proved that whenever this diophantine equation does not have a solution for $x$ then $x^2+(x+1)^2$ is prime. It is conjectured also by Sierpinski that there are infinitely many primes of t …
- 3,866
7
votes
2
answers
396
views
An equation involving perfect numbers
Let $s,x_1,x_2,\cdots, x_s$ be natural numbers not neccesarily distinct.
I am interested in solving the equation $$(x_1+x_2+\cdots +x_s)^s=2^s(x_1\cdot x_2\cdots x_s)^2$$
Some Notes:
I have found t …
- 3,866
31
votes
5
answers
8k
views
Fermat's proof for $x^3-y^2=2$
Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original p …
- 3,866
7
votes
Accepted
A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$
My opinion is:
Step 1.
Repeat the arguments you use in your proofs again and again.
If you are still sure that you have some new (valid) things to say about the equation then go to
Step 2.
Learn how t …