Wall-Sun-Sun Prime
المؤلف:
Sloane, N. J. A
المصدر:
Sequences A113650 and A113651 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
31-8-2020
939
Wall-Sun-Sun Prime
Let 
be the 
th Fibonacci number, and let 
be a Legendre symbol so that
![e_p=(p/5)=<span style=]() {1 for p=1,4 (mod 5); -1 for p=2,3 (mod 5). " src="https://mathworld.wolfram.com/images/equations/Wall-Sun-SunPrime/NumberedEquation1.gif" /> |
(1)
|
A prime 
is called a Wall-Sun-Sun prime if
 |
(2)
|
The first few values of 
are 2, 3, 5, 21, 55, 39, 272, 57, ... (OEIS A113650).
There are no Wall-Sun-Sun primes up to a limit of 
(McIntosh 2004), subsequently extended to 
by PrimeGrid as of November 2015.
Let 
be the 
th Lucas number. Then
 |
(3)
|
for all primes 
. The first few values of 
are 2, 7, 0, 47, 2, 167, 287, ... (OEIS A113651).
Let 
be prime. Then the following are equivalent:
1. 
,
2. 
,
3. 
.
REFERENCES:
McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. https://www.loria.fr/~zimmerma/records/Wieferich.status.
PrimeGrid PRPNet. "Wall-Sun-Sun Prime Search." https://prpnet.primegrid.com:13001.
Sloane, N. J. A. Sequences A113650 and A113651 in "The On-Line Encyclopedia of Integer Sequences."
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