Intereting Posts

Convergence in $L^p$ norm implies pointwise convergence almost everywhere?
What Euclidean functions can the ring of integers be endowed with?
Concecutive last zeroes in expansion of $100!$
Finding $\lim\limits_{n \to \infty} \sum\limits_{k=0}^n { n \choose k}^{-1}$
Is there a bijection from to R?
Proof that a polynomial is irreducible for all $n \ne 4$.
Fourier transform of Schrödinger kernel: how to compute it?
Deriving the sub-differential of the nuclear norm
AM-GM-HM Triplets
Area of a spherical triangle
Showing a function of two variables is measurable
Why cellular maps induce maps of chain complexes?
Prove that prime ideals of a finite ring are maximal
Showing that ${\rm E}=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable
The definition of the logarithm.

Find all pairs of integers $(x, y)$ such that

$$x^3+(x+1)^3+ \dots + (x+15)^3=y^3$$

What I have tried so far:

The coefficient of $x^3$ is $16$ in the left hand side. It is not useful then to trying bound LHS between, for example, $(ax+b)^3$ and $(ax+c)^3$ and then say that $ax+b<y<ax+c$.

- Congruence properties of $x_1^6+x_2^6+x_3^6+x_4^6+x_5^6 = z^6$?
- Triples of positive integers $a,b,c$ with rational $\sqrt{\frac{c-a}{c+b}},\sqrt{\frac{c+a}{c-b}}$
- Solve $x^3 +1 = 2y^3$
- Finding solutions to the diophantine equation $7^a=3^b+100$
- Diophatine equation $x^2+y^2+z^2=t^2$
- Integer solutions of $x^4 + 16x^2y^2 + y^4 = z^2$

I also tried to use modulo a prime. But it seems unlikely to bound variables this way.

EDIT : Though, it can be factored as $(2x+15)(x^2+15x+120)=(y/2)^3$. LSH factors are almost co-prime and we can say that $x^2+15x+120=3z^3$ or $x^2+15x+120=5z^3$. These are still too difficult to solve!

Any ideas?

- Quintic diophantine equation
- How to express the function $\mathbb{N} \to \mathbb{N}\times \mathbb{N}$ as a mathematical statement?
- How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$
- Modular Arithmetic question, possibly involving Chinese remainder theorem
- Partition of ${1, 2, … , n}$ into subsets with equal sums.
- Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?
- How many ways are there to shake hands?
- $\sqrt{13a^2+b^2}$ and $\sqrt{a^2+13b^2}$ cannot be simultaneously rational
- bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$
- Proof Check Lemma 2.2.10 in Tao

This isn’t a complete solution, but I hope it gives you an approach. (It’s too long for a comment)

Since we have $$\sum_{r=1}^{n} r^3=\left(\frac { n(n+1)}{2}\right)^2$$

You can write

\begin{align}

x^3+(x+1)^3+ \dots + (x+15)^3

&=\sum_{r=1}^{x+15} r^3-\sum_{r=1}^{x-1} r^3

\\

&=\left(\frac { (x+15)(x+16)}{2}\right)^2-\left(\frac { x(x-1)}{2}\right)^2\\

&=\left[\left(\frac { (x+15)(x+16)}{2}\right)-\left(\frac { x(x-1)}{2}\right)\right]\left[\left(\frac { (x+15)(x+16)}{2}\right)+\left(\frac { x(x-1)}{2}\right)\right]\\

\end{align}

Simplifying this, we get $$(x^2+15x+120)(2x+15)=\left(\frac y2 \right)^3$$

You are looking for $(x, y) \in \mathbb{Z}^2$ for which

$$ \sum_{i= 0}^{15} (x+i)^3 = y^3, \tag{0}$$

that is,

$$ \sum_{i= 0}^{15} \left( x^3 + 3 i x^2 + 3i^2 x + i^3 \right) = y^3, $$

that is,

$$ 16 x^3 + 3 \frac{15 (15+1)}{2} x^2 + 3 \frac{15 (15+1)(2 \times 15 + 1)}{6} + \left( \frac{15 (15+1)}{2} \right)^2 = y^3, $$

that is,

$$ 16 x^3 + 360 x^2 + 3720 x + 14400 = y^3, $$

which can be written as

$$ 8 (2x^3 + 45 x^2 + 465 x + 1800) = y^3 $$

Can you get anywhere from here?

- When are two simple tensors $m' \otimes n'$ and $m \otimes n$ equal? (tensor product over modules)
- How to partial differentiate a total differential and be rigorous on all the notion?
- Generalised inclusion-exclusion principle
- What does it mean when two functions are “orthogonal”, why is it important?
- How to evaluate $\xi(0)$?
- Closed subspace of $l^\infty$
- How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 6?
- Analogy of ideals with Normal subgroups in groups.
- How can this English sentence be translated into a logical expression?
- Summing digits of powers of 2 to get 1 2 4 8 7 5 pattern
- Given an invertible matrix $A$ such that all elements in $A$ and in $𝐴^{−1}$ are integers, find $|𝐴^4|$
- Use Mathematical Induction to prove that $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} +…+\frac{1}{n(n+1)}=1-\frac{1}{n+1}$
- Chased By a Lion and other Pursuit Problems
- How to do well on Math Olympiads
- Stolz-Cesaro Theorem, 0/0 Case