18
votes

Accepted

### Can the Pythagorean Graph be finitely colored?

This paper shows that the chromatic number is infinite.
Indeed, Theorem 1.1 part (i) with $a=b=c=1$ is what you want.

11
votes

Accepted

### Suitable closed form for the A079501

Yes!
$i$ is the first part in the composition
$j + 1$ is the number of other parts in the composition
$+1$ accounts for the case that there is only one part
Summing over $i$ and $j$, we want to ...

11
votes

Accepted

### Integrality of a quotient of Fermat numbers

The fraction equals
$$\prod_{k=0}^{n-1}\frac{2^{2^m-2^k}-1}{2^{2^n-2^k}-1},$$
hence it suffices to show that
$$\prod_{k=0}^{n-1}\frac{x^{2^m-2^k}-1}{x^{2^n-2^k}-1}\in\mathbb{Z}[x].$$
The numerator and ...

8
votes

Accepted

### Fibonacci and matrix modular exponentiation

For Q1, you may like to check Fiduccia algorithm and a more recent paper A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence.
This is also applicable to Q2 (...

8
votes

### Subset of $\mathbb N$ missing at least a class modulo each prime

If $S=\{p_1,p_1p_2,p_1p_2p_3,\cdots\}$ where $p_n$ is the $n^\text{th}$ prime, then $\operatorname{card}(S\operatorname{mod}p_n)\le n\lt p_n$ for all $n$.

4
votes

Accepted

### heights of ideal classes and reduction theory for Bhargava cubes

No, it is not true that $M(D) = o(|D|^{3/2})$. For example,
if $D = -4abcd$ where $a,b,c,d$ are
odd, pairwise coprime and of roughly equal size, then
the forms $ab\,x^2+cd\,y^2, ac\,x^2+bd\,y^2, ad\,...

4
votes

### An equality between $\pi$ and $\Gamma$ function

Your series is a special case of Gauss' hypergeometric series. In standard notation, it is
$${}_2F_1\left(\begin{matrix}1/2,1/2\\2\end{matrix};-1\right).$$
In standard tables (e.g. 15.4.26 in https://...

3
votes

### Unique "clique" of differences in $\mathbb{Z}/m\mathbb{Z}$

There is no such pair of $\epsilon$ and $N$: Set $m=k^2$ with $k$ big enough, $G=\mathbb Z/k^2\mathbb Z$ and $H=k\mathbb Z/k^2\mathbb Z$. With $A=H\setminus\{0\}$ we have
\begin{equation}
(A+x)\cap(A+...

2
votes

Accepted

### Recursion for the Chebyshev transform of $m^n$

UPDATED. The argument below is corrected.
Apparently, under Chebyshev transform of a generating function $A(x)$ OP understands a function $B(x) := C(-x^2)A(xC(-x^2))$, where $C(x):=\frac{1-\sqrt{1-4x}}...

2
votes

### Refinement of a theorem of Koblitz-Ogus

We need the result you mention in work we're currently doing together with Heidi Goodson and my PhD student Andrea Gallese. We've also been unable to locate this precise result anywhere in Kubert's ...

2
votes

### Proofs of the Chevalley-Warning Theorem

Whether the following proof is different from the other ones already mentioned could be debated, but to me it feels different enough to be worth mentioning separately. As noted in Gjergji Zaimi's ...

1
vote

### Could I possibly exploit distinct odd primes raised to 6 to solve Exact Three Cover, when reducing it in Subset Sum?

Going by your notation, $c \in C$, $c=(x_1, x_2, x_3), x_i \in S \simeq \{1, \cdots , N \}$, and $\phi(c) \to \mathbb N$ the map you mentioned, $(\alpha_1,\cdots, \alpha_3) \mapsto \sum_i p_{\alpha_i}...

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