论文标题
$ \ frac {1} {2}（f_n-1）的表示（f_ {n+1} -1）$和$ \ frac {1} {1} {2} {2}（f_n-1）（f_ {n+2} -1）$
Representation of $\frac{1}{2}(F_n-1)(F_{n+1}-1)$ and $\frac{1}{2}(F_n-1)(F_{n+2}-1)$
论文作者
论文摘要
令$ a，b \ in \ mathbb {n} $相对典型。我们考虑$（a-1）（b-1）/2 $，这是在研究$ pq $ - th cyclotomic多项式的研究中，其中$ p，q $是不同的素数。我们证明$（A-1）（B-1）/2 $的两种可能表示为$ a $ a $和$ b $的非负线线性组合。令人惊讶的是，对于每对$（a，b）$，两个表示中只有一个，并且该表示也是唯一的。然后，我们研究$（f_n-1）（f_ {n+1} -1）/2 $和$（f_n-1）（f_n-1）（f_ {n+2} -1）/2 $的表示形式，其中$ f_i $是$ i^{th} $ i^{th} $ fibonacci编号，并观察几个不错的模式。
Let $a, b\in \mathbb{N}$ be relatively prime. We consider $(a-1)(b-1)/2$, which arises in the study of the $pq$-th cyclotomic polynomial, where $p,q$ are distinct primes. We prove two possible representations of $(a-1)(b-1)/2$ as nonnegative, integral linear combinations of $a$ and $b$. Surprisingly, for each pair $(a,b)$, only one of the two representations exists and the representation is also unique. We then investigate the representations of $(F_n-1)(F_{n+1}-1)/2$ and $(F_n-1)(F_{n+2}-1)/2$, where $F_i$ is the $i^{th}$ Fibonacci number, and observe several nice patterns.
