Intereting Posts

Is there anything like “cubic formula”?
Forming Partial Fractions
Is T an infinity spectrum whenever T is a spectrum?
Why are polynomials defined to be “formal”?
Pythagorean Theorem Proof Without Words 6
How can one write $z^{-1}$ as a Stieltjes function?
On the propagation of singularities in PDE
relative size of most factors of semiprimes, close?
regularization of a divergent integral
Solve the equation $(2^m-1) = (2^n-1)k^2$
Game about placing pennies on table
For a family of sets $\mathbb{U}$, $\cup_{arbitrary}(\cap_{finite} U)$ $\forall U \in \mathbb{U}$ is stable under $\cap_{finite}$.
prove: coefficients of $f(x)$ are rational numbers
Planar Realization of a Graph in Three-Space
Length of Chord is Independent of Point P

**Prove that $2^n +1$ in never a perfect cube**

I’ve been thinking about this problem, but I don’t know how to do it. I know that if $m^3=2^n+1$, then $m$ should be an odd number, but I ‘m not able to get to a contradiction.

- Solution of $ax=a^x$
- Is the discriminant of a second order equation related to the graph of $ax^2+bxy+cy^2+dx+ey+f=0$?
- Real domain and range function to find all functions with nonzero x.
- proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$
- How do I solve $2^x + x = n$ equation for $x$?
- Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$

- Evaluation of the sum $\sum_{i=1}^{\lfloor na \rfloor} \left \lfloor ia \right \rfloor $
- 31,331,3331, 33331,333331,3333331,33333331 are prime
- Integer ordered pairs $(x,y)$ for which $x^2-y!$…
- Prove that $x^{2} \equiv -1$ (mod $p$) has no solutions if prime $p \equiv 3\pmod 4$.
- Proof strategy - Stirling numbers formula
- Generating Coprime Integers
- Determine polynomial whose roots are a linear combination of roots of another polynomial
- Highest power of a prime $p$ dividing $N!$
- Last non Zero digit of a Factorial
- How to solve this quadratic congruence equation

Suppose that $2^n+1=m^3$. Then

$$2^n=m^3-1=(m-1)(m^2+m+1),$$

so each of $m-1$ and $m^2+m+1$ is a power of $2$.

But $m^2+m+1$ is odd, and therefore $m^2+m+1=1$. That forces $m^2+m=0$, giving $m=0$ (impossible) or $m=-1$ (also impossible).

Andre’s answer is neat, I’ve got a similar solution. Assume $m^3 = 2^n + 1$ then $m$ is odd therefore $m=2p+1$ thus $$8p^3 + 12p^2 + 6p + 1 = 2^n + 1 \implies 2p(4p^2 + 6p + 3) = 2^n$$ Note that $2p$ **and** $4p^2 + 6p + 3$ must be powers of $2$, however the latter is odd. Then, again $4p^2 + 6p + 3=1 \implies p=-1 \text{ or } p=-1/2$ both of which are impossible because $m > 0$

- trigonometry query
- Show that every graph $G$ has a bipartite subgraph with at least half of the edges of $G$
- Question on problem: Equivalence of two metrics $\iff$ same convergent sequences
- Let $f$ be an analytic isomorphism on the unit disc $D$, find the area of $f(D)$
- Why rationalize the denominator?
- Evaluate the definite integral $ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^4 +1} \ \ dx $
- Questions on scheme morphisms
- Why hasn't GCH become a standard axiom of ZFC?
- Does this characterize compactness?
- Does weak compactness imply boundedness in a normed vector space (not necessarily complete)?
- An application of Hadamard-Lévy's theorem
- The diffential of commutator map in a Lie group
- Importance of Least Upper Bound Property of $\mathbb{R}$
- Count $k$-subsets with at least $d>1$ different elements (pairwise)
- Is there an automorphism of symmetric group of degree 6 sending a transposition to product of two transpositions?