Intereting Posts

Product rule for logarithms works on any non-zero value?
Negation of uniform convergence
Is every compact subset of $\Bbb{R}$ the support of some Borel measure?
Cube of harmonic mean
Different arrows in set theory: $\rightarrow$ and $\mapsto$
The integral of a closed form along a closed curve is proportional to its winding number
Distribute $N$ objects to $K$ boxes such that no box has more than $c$ objects in it
Deriving the formula of the Surface area of a sphere
Probability distribution and their related distributions
iterated dual vector spaces
If $f\in L^1(\mathbb{R})$, then $\sum_{n\ge 1}f(x+n)$ Converges for a.e. $x$.
Prove that there are infinitely many primes of the form $8k + 3$
Real Dirichlet characters mod $k$
Showing an ideal is prime in polynomial ring
Simplify a combinatorial sum

Take a given Diophantine equation. Chances are, we can’t find any solutions. But if it’s an equation of a certain form, we may get lucky and may be able not only to find a solution, but be able to classify all the solutions of such an equation.

I’m preparing a talk on Diophantine Equations. I’m interested in exploring how the difficulty of Diophantine equations increase as both the number of variables and the degree of the equation increases. Of course, due to Matiyasevich’s resolution of Hilbert’s Tenth Problem, we know that no general algorithm to determine whether or not a given Diophantine Equation has a solution.

Here’s my current understanding of the “frontier” of Diophantine Equations:

- Why does the diophantine equation $x^2+x+1=7^y$ have no integer solutions?
- Linear Diophantine Equations in Three Variables
- $\frac{x^5-y^5}{x-y}=p$,give what p ,the diophantine equation is solvable
- Equation over the integers
- Birational Equivalence of Diophantine Equations and Elliptic Curves
- Generalized Pell's Equation: Why is 4 special?

**Variables $=1$, Degree $=n$:**

The solutions are completely determined by the Rational Root Theorem. Once you have all the rational roots, just look for the roots that are also integers.

**Variables $=k$, Degree $=1$:**

The equation is of the form $$a_1 x_1 + … + a_k x_k = d.$$

This has solutions if and only if the greatest common divisor of $a_1,…,a_k$ divides $d$, and in that case, we have a nice formula parametrising the solutions (which I won’t repeat here).

**Variables $=2$, Degree $=2$:**

There exist solutions to an equation $P(x,y)$ if and only if we have:

1) There exist solutions to $P(x,y) \mod n$ for all integers $n$ (so in practise, you just need to verify there exists solutions $\mod p$ for all primes $p$, and then by Hensel’s Lemma, and the Chinese Remainder Theorem, we have solutions for all $n$.

2) There exist real solutions to $P(x,y)$

If we can find one solution, we can find all rational solutions through stereographic projection. Once you have all rational solutions, look at the solutions where the denominator divides the numerator to get all the integer solutions.

**Variables $=2$, Degree $=3$:**

This is handled by elliptic curves. Broadly speaking, provided that the equation is “nice enough”, the set of solutions to the Diophantine equation has an abelian group structure. This consists of a torsion free part, which can be one of 16 possible groups, and a torsion free part, which looks like $\mathbb{Z}^r$ for some natural number $r$. The $r$ here is called the rank of an elliptic curve, and computing it is, in general, computationally difficult.

**Variables $\geq 2$, Degree $\geq 4$:**

In general, this is no man’s land. Of course, some equations we may have luck with (take $x^4 + y^4 = z^4$; there is an elementary proof that this has no solutions in the natural numbers), but in general, you’re probably out of luck.

The other two types of Diophantine equations that I haven’t spoken about are exponential Diophantine equations (such as Catalan’s conjecture: $x^a – y^b =1$ only has one solutions, namely $x=3, a = 2, y=2, b=3$), and infinite Diophantine equations. These are even more intractable, so I won’t touch on these at all.

My question is, in my above analysis, have I missed out anything obvious/is there anything glaringly wrong?

- Would this proof strategy work for proving the lonely runner conjecture?
- Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$
- $(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is
- Sailors, monkey and coconuts
- Solving the equation $x^{2}-7^{y}=2$
- Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational?
- Can multiplication be defined in terms of divisibility?
- Proving $\mathbb{Z}$, $\mathbb{Z}$, $\mathbb{Z}$, and $\mathbb{Z}$ are euclidean.
- Continued fraction of a square root
- Prime Divisors of $x^2 + 1$

- If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges?
- Limit comparison test proof
- How far is being star compact from being countably compact？
- Induction: show that $\sum\limits_{k=1}^n \frac{1}{\sqrt{k}} < 2 \sqrt{n}$ for all n $\in Z_+$
- Recurrence telescoping $T(n) = T(n-1) + 1/n$ and $T(n) = T(n-1) + \log n$
- The interval andd are equivalent. Is my proof correct?
- Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.
- Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$
- Is $ d(X) \le s(X)? $
- system of matrix equations
- Are all the norms of $L^p$ space equivalent?
- Measure Spaces: Uniform & Integral Convergence
- Why does $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$?
- Is the opposite of the Second Derivative Test also true?
- Finding a unique subfield of $\mathbb{Q}(\zeta)$ of degree $2$?