Intereting Posts

What is the image near the essential singularity of z sin(1/z)?
Intuitively, how should I think of Measurable Functions?
Trees whose complement is also a tree
proving the function $\frac{1}{1+x^2}$ is analytic
Generalised Binomial Theorem Intuition
If every cyclic subgroup of $G$ is normal so is every subgroup?
Equivalence of categories ($c^*$ algebras <-> topological spaces)
Finite Abelian groups: $G \times H \cong G\times K$ then $H\cong K$
What is $\Bbb{R}^n$?
How (and why) would I reparameterize a curve in terms of arclength?
Is $ (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D)$ true for all sets $A, B, C$ and $D$?
Showing that $X_{1:n}$ is sufficient for $\eta$, by factorization
Is there an flat unordered pairing function in ZFC?
Transitive subgroup of symmetric group
Binomial Distribution Problem – Airline Overbooking

I am trying to prove that $x^3+y^4=7$ has no integer solutions, but i have no idea how to start, please helps. I have tried to consider mod 7 to restrict the number of possible $x^3$ because $x^3 \equiv -1,0,1 \pmod{7}$, but it is not working.

- What is wrong with this effort to generalize Bertrand's Postulate using the Inclusion-Exclusion Principle
- Multivariable Equation: $4ab=5(a+b)$
- Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers.
- How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?
- Need help understanding Erdős' proof about divergence of $\sum\frac1p$
- $\Bbb Z_m \times \Bbb Z_n$ isomorphic to $\Bbb Z_{\operatorname{lcm}(m,n)}\times \Bbb Z_{\gcd(m,n)}$

- $a, b \in\Bbb N$, $A=\sqrt{a^2+2b+1}+\sqrt{b^2+2a+1}\in \Bbb N $, to show that $a = b$.
- Common denominators
- Solve in $\mathbb Z^3$.
- Prove that $371\cdots 1$ is not prime
- Foundational proof for Mersenne primes
- An Algorithm to compute the GCD of polynomials of coprime numbers?
- Find the number of positive integers solutions of the equation $3x+2y=37$
- Solving $\phi(n)=84$

Consider the equation modulo $13$. Then $x^3$ can be $0,1,5,8,12$ and $y^4$ can be $0,1,3,9$. None of these add to $7$ modulo $13$.

I chose $13$ because $3|\phi(13)$ and $4|\phi(13)$, so I could get restrictions on both $x^3$ and $y^4$.

universalset’s answer is right, in these problems a useful heuristic is to find a number that produces remainders such that both sides cannot become equal. Here is a short Python program to find such numbers for this problem.

```
for m in range(2,100): #We hope to find such number less than 100
a= set([x**3%m for x in range(0,1000)]) #Hoping remainders will cycle somewhere less that 1000
b= set([y**4%m for y in range(0,1000)]) #Hoping remainders will cycle somewhere less that 1000
flag = 0;
for x in a:
if (7%m+x%m)%m in b:
flag=1;
break;
if flag==0:
print (m,a,b)
```

- What are “set-theoretic maps”?
- Hard Integral $\frac{1}{(1+x^2+y^2+z^2)^2}$
- Binomial Theorem Proof by Induction
- Proving $f$ is Lebesgue integrable iff $|f|$ is Lebesgue integrable.
- An inequality $\,\, (1+1/n)^n<3-1/n \,$using mathematical induction
- $X$ is homeomorphic to $X\times X$ (TIFR GS $2014$)
- Dimension of the solution of a second order homogenous ODE
- Probability of 3 Heads in 10 Coin Flips
- What is the general term for concepts like length, area and volume?
- Connected But Not Path-Connected?
- How to extend this extension of tetration?
- Prove that $\text{nil}(\overline{R})=\sqrt{A}/A$
- necessary and sufficient condition for the Poisson's equation to admit a solution$?$
- Convergence rate of Newton's method
- Categorification of characteristic polynomial