# Tag Info

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.
• 985
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 ...
• 726
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 ...
• 99.1k
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 (...
• 30.6k

### 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 $n$.
• 11.9k
Accepted

• 30.6k

### 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 ...
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}...