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It was shown previously that the number 1 can be written as the sum and difference of at most three different tetrahedral numbers infinitely many different ways. But the dodecahedral numbers are a sub …

Prove that $\sum_{k=0}^n {k+1\choose 2}^R + \sum_{k=0}^{-n-2} {k+1\choose 2}^R = 0.$ This can be shown using Faulhaber's formula but it's very long. Is there a nicer, shorter method? Any thoughts or i …

Here's the problem: start with $2^n$, then take away $\frac{1}{2}a^2+\frac{3}{2}a$ starting with $a=1$, and running up to $a = 2^{n+1}-2$, evaluating modulo $2^n$. Does the resulting sequence contain …

Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$? That is, if there are infinitely many different $a$ and for which at least one value of …

I want to show if it's true that $60m^2+6m-1$ is a quadratic residue modulo $6gm+1$ for all $m \in \mathbb{N}$ and $6gm+1$ is prime, for infinitely many positive integers $g$. (I'm not 100% certain th …

Like the title asks, are there any known positive integers $k$ such that there are infinitely many primes that can be written as a sum of $k$ fibonacci numbers? For $k=1$ this is a famous unsolved pro …

The original proof of the four square theorem relied on Euler's great identity that the sum of four squares times the sum of four squares equals the sum of four squares. Currently it is an open proble …

In EM Wright's original paper, AN EASIER WARING'S PROBLEM (1934), he presents two nontrivial identities. The first is $(x+4)^4-2(x+3)^4+2(x+1)^4-x^4=48x+96$ and the second is $(x+3)^5-2(x+2)^5+x^5+(x- …

Prove for the Bernoulli numbers $B_n$, that for all $a \in \mathbb{N}$, that $ \sum_{i=0}^{2a+1} {2a+1 \choose i} B_{2a+1-i} [ (n+1)^i+(-n)^i ] =0 $. As much as this is a neat identity, it's a crucial …

It's well known that $1$ is the sum of three cubes infinitely many different ways but is it true for perhaps the tetrahedral numbers as well? Let $T_n = (1/6)n(n+1)(n+2)$. Then the following are the f …

Consider how the iterated sum $ \sum_{k_{j-1}=0}^{k_j}....\sum_{k_1=0}^{k_2} \sum_{k=0}^{k_1} 1 $ produces the diagonals on pascal's triangle, so it has a nice closed form. Does the iterated sum $ \su …

I'm dealing with the expression $x^2+y^2+6z^2+8xy+4x+4y−6xz−6yz$. I want to show that this expression is always non-zero whenever $x,y$ and $z$ are positive integers. How does one do this? (Note that …

I'm looking at the following generalizations for sums of two cubes. $u^3+v^3=(u+v)(u^2-uv+v^2).$ $u^3+(u+r)^3+v^3+(v+r)^3=(u+v+r)(2u^2-2uv+2v^2+ru+rv+2r^2).$ $u^3+(u+q)^3+(u+r)^3+(u+q+r)^3+v^3+(v+q)^3 …

Prove that $p\mid\genfrac[]0{}{p^w}k$ where $p$ is an odd prime, $w \in \mathbb{N}$, $1<k<p^w$ and $k \neq p^v$ for some positive integer $v<w$. This has to be already done I just can't find where. Wh …

Let $T_n = \frac{1}{6}n(n+1)(n+2)$ denote the $n$th Tetrahedral number. The first several solutions to squares as sums of two Tetrahedral numbers are {T_n,T_m,a^2} 1 5 6\ 1 8 11\ 1 22 45\ 1 24 51\ 1 6 …