1
vote
0answers
130 views
Diophantine: a^n + b^n + c^n = d^n and a^n + b^n = c^n + d^n
Let us consider the equation $a^n+b^n=c^n$ for positive integers $a,b,c$ and $n\ge 2$. The $n=2$ case has a well-known and beautiful parametrization known as Pythagorean triples. F …
13
votes
1answer
263 views
Is there an online encyclopedia of Diophantine equations (OEDE)?
Hello all!
I'm just wondering if there is an online encyclopedia of Diophantine equations (OEDE), analogous to the OEIS for sequences.
While trying to solve one Diophantine equat …
0
votes
2answers
98 views
Classification of these Binary Quadratic Forms
What are necessary and sufficient conditions on a binary quadratic form $ax^2+bxy+cy^2$, with integer coefficients and solution set in integers, to be equivalent to $x^2-y^2$, and …
1
vote
1answer
553 views
The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus ?
After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial.
In partic …
5
votes
3answers
420 views
Can repunits be perfect cubes?
Is it true that the equation $10^{n}-9m^{3}=1$ has only one positive integer solution, namely $n=m=1$? I can't find the answer. This has an equivalent description that the repunits …
9
votes
1answer
695 views
Effect of abc conjecture on Fermat’s Last Theorem
A website ( http://www.math.unicaen.fr/~nitaj/abc.html#Consequences ) says that the $abc$ conjecture implies that there are only finitely many solutions to the equation $x^n+y^n=z^ …
2
votes
2answers
208 views
Help with this system of Diophantine equations
A couple hours ago, I'd posted a Diophantine equation question, but realized that I'd committed a rather preposterous blunder deriving it.
This is the actual question which I'm tr …
1
vote
3answers
283 views
Help with this Diophantine equation
Note: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at http://mathoverflow.net/questions/131353/help-with-this-sys …
3
votes
1answer
156 views
Diophantine equation with primitive nth root of unity
Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\x …
2
votes
1answer
454 views
What is known about a^2 + b^2 = c^2 + d^2
Could you state or direct me to results regarding the Diophantine equation $a^2+b^2=c^2+d^2$ over integers? Specifically, I am looking for a complete parametrization. In the case t …
7
votes
2answers
226 views
A sequence based on Catalan–Mihăilescu problem
It was conjectured by Catalan in 1844 that the only solutions of the equation $x^a-y^b=1$ over variables $a,b,x,y\in\mathbb{N^+}$ are trivial ones: $3^1-2^1=1$ and $3^2-2^3=1$. The …
0
votes
0answers
117 views
Reference for original paper (but translated to English) of Matiyasevich’s proof of Fibonacci relation being Diophantine?
Hello. I am a maths undergraduate. I am doing a project about history of mathematics. I am looking for the original solution to Hilbert's 10th problem, or at least the theorems tha …
16
votes
1answer
533 views
Solutions to $\binom{n}{5} = 2 \binom{m}{5}$
In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says:
On National Public Radio, the Weekend Edition program posed the
following probability problem: Given a c …
6
votes
3answers
274 views
References on techniques for solving equations with discontinuous functions such as floor and ceiling?
Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm …
4
votes
1answer
247 views
What analytic tools can provide a lower bound for this Diophantine equation?
The resolution of the Diophantine equation $$m! = n(n+1)$$ was asked on M.SE. My intuition says that this cannot be solved by elementary means - apologies if I am mistaken.
I felt …

