# In number theory what do we call Lucas's theorem?

We have three important results by Lucas :

1- Let $q,a\in \mathbb{N}$ and both are $>1$ such as : $a^{q-1}\equiv1\ [q]$ and for all prime divisors $p$ of $q-1$, $a^{\frac{q-1}{p}}\not\equiv 1 \ [q]$. Then $q$ is prime and $a$ is a generator of $\mathbb{F}_q$.

2- Let $p$ a prime number, $m,n$ two positive integers. If $m=m_0+m_1p+...+m_{k}p^{k}$ and $n=m_0+m_1p+...+m_{k}p^{k}$ then $\binom{m}{n}\equiv\prod\limits_{i=0}^{k}\binom{m_i}{n_i}\ [p]$.

3- Let $(F_n)_{n\in\mathbb{N}}$ the Fibonacci sequence. Let $F_0=0$, $F_1=1$ and $F_{n+2}=F_{n+1}+F_n$. Then $\gcd(F_n,F_m)=F_{\gcd(n,m)}$.

In general, what result is known as "Lucas's theorem" ?

I found that the 1 is a theorem by Lucas in 1876 for primality so maybe it's more appropriate to say that it's "Lucas's test" then the 2 is The Lucas's theorem published in 1878 and the 3 represents the Lucas's sequence which generalizes Fibonacci and a consequence is the formula for the $\gcd$ but I ignore the year of appearance for this one...

Thanks in advance !

• Unqualified reference to a "Lucas theorem" would be meant as a reference to no.2. But this question in its current form is opinion based. – sdf Sep 15 '16 at 16:52
• According to the Wikipedia article "Lucas's Theorem" it's #2, See also the wiki article on Edouard Lucas. – DanielWainfleet Sep 15 '16 at 17:18