--- In

primenumbers@yahoogroups.com, "paulmillscv" <paulmillscv@y...> wrote:

> Dear Professor Chris Caldwell and all Primers,

> Hello. Greetings from England to all my friends. The F.I.D.N

> Christmas quiz time is fast approaching and as usual I have some

> difficult questions for you all. To get you in the mood can you

> find a solution to X^7 == 31 (mod p) I.e. find X an integer and p

> a prime which satisfy this modular equation.

Oooh, a puzzle!

Let's see... if I can read what I've quickly scribbled on this napkin, p is

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002660001610073694271481942089924707468045727786571849791328008691648699739525329796838119965958195130176131768771278250667488356000909769602624205809244992753070304972931405217590727680545352971103551730525130866411254246473378605300816282734372995535289681239925398549901960474742174432203645452861369827704969342952978182043153348607189796689003179104506131436897643525897165715077590444519639378572831732088417753060825002419044585794676788518688051676202824708997859659835305791460526322955181528366987832170988448802894797910003061765609881663165410613063630934264116117379171463080169854184309031440200335675149583070885689193098243769734862642705454835478053736482873427728485437909085167204835144884454146855555960143321081043003558440501905685432453215652020178273836809427955916350633721823042430063012210351478908640935576501429129128271364989511404641

Finding X is left as an exercise for the reader.

R.A. Twain