Existence Wall–Sun–Sun prime

in study of pisano period



k
(
p
)


{\displaystyle k(p)}

, donald dines wall determined there no wall–sun–sun primes less



10000


{\displaystyle 10000}

. in 1960, wrote:

the perplexing problem have met in study concerns hypothesis



k
(

p

2


)
≠
k
(
p
)


{\displaystyle k(p^{2})\neq k(p)}

. have run test on digital computer shows



k
(

p

2


)
≠
k
(
p
)


{\displaystyle k(p^{2})\neq k(p)}





p


{\displaystyle p}





10000


{\displaystyle 10000}

; however, cannot prove



k
(

p

2


)
=
k
(
p
)


{\displaystyle k(p^{2})=k(p)}

impossible. question closely related one, can number



x


{\displaystyle x}

have same order mod



p


{\displaystyle p}

, mod




p

2




{\displaystyle p^{2}}

? , rare cases give affirmative answer (e.g.,



x
=
3
,
p
=
11


{\displaystyle x=3,p=11}

;



x
=
2
,
p
=
1093


{\displaystyle x=2,p=1093}

); hence, 1 might conjecture equality may hold exceptional



p


{\displaystyle p}

.

it has since been conjectured there infinitely many wall–sun–sun primes. no wall–sun–sun primes known of april 2016.

in 2007, richard j. mcintosh , eric l. roettger showed if exist, must > 2×10. dorais , klyve extended range 9.7×10 without finding such prime.

in december 2011, search started primegrid project. of april 2016, primegrid has extended search limit 1.9×10 , continues.