1 results
A NOTE ON GENERALISED WALL–SUN–SUN PRIMES
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 3 / December 2023
- Published online by Cambridge University Press:
- 28 February 2023, pp. 373-378
- Print publication:
- December 2023
-
- Article
-
- You have access
- HTML
- Export citation
-
Let a and b be positive integers and let
$\{U_n\}_{n\ge 0}$ be the Lucas sequence of the first kind defined by 
$$ \begin{align*}U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{for }n\ge 2.\end{align*} $$We define an
$(a,b)$-Wall–Sun–Sun prime to be a prime p such that 
$\gcd (p,b)=1$ and 
$\pi (p^2)=\pi (p),$ where 
$\pi (p):=\pi _{(a,b)}(p)$ is the length of the period of 
$\{U_n\}_{n\ge 0}$ modulo p. When 
$(a,b)=(1,1)$, such primes are known in the literature simply as Wall–Sun–Sun primes. In this note, we provide necessary and sufficient conditions such that a prime p dividing 
$a^2+4b$ is an 
$(a,b)$-Wall–Sun–Sun prime.
