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Primality Test - CBZ-100-A
Hi,
I have been working on a deterministic for the last few days. I did not have a proof but Thanks to SM turns out the proof has already been made by by Lagrange in 1775 (at least in part): [URL]http://primes.utm.edu/notes/proofs/MerDiv2.html[/URL] +Related thread link: [URL]http://www.mersenneforum.org/showpost.php?p=486512&postcount=77[/URL] Pari-GP code: [CODE] print("CBZ-110-A Primality test by Rashid Naimi") theOrder = 1 \\\\\ k= 2 \\ theLimit =3 \\\ thePrime = 1; forprime(q=theLimit,theLimit+19^4,{ p=k*(q^theOrder )+1; n = Mod(2,p)^(q^theOrder )-1; if(gcd(lift(n),p)>1, print("\n**** ", k,"*(",q,"^",theOrder,")+1 > ",primeFlag); thePrime =p; primeFlag=isprime(p); print(#digits(thePrime)," dd"); \\next(19); \\ Disactivate to scan the entire range ); }) ## isprime(thePrime ) ## print(#digits(thePrime)," dd") print("CBZ-110-A Primality test by Rashid Naimi") print("Order = ",theOrder) print("k = ",k) [/CODE][CODE] CBZ-110-A Primality test by Rashid Naimi **** 2*(3^1)+1 > 1 1 dd **** 2*(11^1)+1 > 1 2 dd **** 2*(23^1)+1 > 1 2 dd **** 2*(83^1)+1 > 1 3 dd **** 2*(131^1)+1 > 1 3 dd **** 2*(179^1)+1 > 1 3 dd **** 2*(191^1)+1 > 1 3 dd **** 2*(239^1)+1 > 1 3 dd **** 2*(251^1)+1 > 1 3 dd **** 2*(359^1)+1 > 1 3 dd **** 2*(419^1)+1 > 1 3 dd **** 2*(431^1)+1 > 1 3 dd **** 2*(443^1)+1 > 1 3 dd **** 2*(491^1)+1 > 1 3 dd **** 2*(659^1)+1 > 1 4 dd **** 2*(683^1)+1 > 1 4 dd **** 2*(719^1)+1 > 1 4 dd **** 2*(743^1)+1 > 1 4 dd **** 2*(911^1)+1 > 1 4 dd **** 2*(1019^1)+1 > 1 4 dd **** 2*(1031^1)+1 > 1 4 dd **** 2*(1103^1)+1 > 1 4 dd **** 2*(1223^1)+1 > 1 4 dd **** 2*(1439^1)+1 > 1 4 dd **** 2*(1451^1)+1 > 1 4 dd **** 2*(1499^1)+1 > 1 4 dd **** 2*(1511^1)+1 > 1 4 dd **** 2*(1559^1)+1 > 1 4 dd **** 2*(1583^1)+1 > 1 4 dd **** 2*(1811^1)+1 > 1 4 dd **** 2*(1931^1)+1 > 1 4 dd **** 2*(2003^1)+1 > 1 4 dd **** 2*(2039^1)+1 > 1 4 dd **** 2*(2063^1)+1 > 1 4 dd **** 2*(2339^1)+1 > 1 4 dd **** 2*(2351^1)+1 > 1 4 dd **** 2*(2399^1)+1 > 1 4 dd **** 2*(2459^1)+1 > 1 4 dd **** 2*(2543^1)+1 > 1 4 dd **** 2*(2699^1)+1 > 1 4 dd **** 2*(2819^1)+1 > 1 4 dd **** 2*(2903^1)+1 > 1 4 dd **** 2*(2939^1)+1 > 1 4 dd **** 2*(2963^1)+1 > 1 4 dd **** 2*(3023^1)+1 > 1 4 dd **** 2*(3299^1)+1 > 1 4 dd **** 2*(3359^1)+1 > 1 4 dd **** 2*(3491^1)+1 > 1 4 dd **** 2*(3539^1)+1 > 1 4 dd **** 2*(3623^1)+1 > 1 4 dd **** 2*(3779^1)+1 > 1 4 dd **** 2*(3803^1)+1 > 1 4 dd **** 2*(3851^1)+1 > 1 4 dd **** 2*(3863^1)+1 > 1 4 dd **** 2*(3911^1)+1 > 1 4 dd **** 2*(4019^1)+1 > 1 4 dd **** 2*(4211^1)+1 > 1 4 dd **** 2*(4271^1)+1 > 1 4 dd **** 2*(4391^1)+1 > 1 4 dd **** 2*(4871^1)+1 > 1 4 dd **** 2*(4919^1)+1 > 1 4 dd **** 2*(4943^1)+1 > 1 4 dd **** 2*(5003^1)+1 > 1 5 dd **** 2*(5039^1)+1 > 1 5 dd **** 2*(5051^1)+1 > 1 5 dd **** 2*(5171^1)+1 > 1 5 dd **** 2*(5231^1)+1 > 1 5 dd **** 2*(5279^1)+1 > 1 5 dd **** 2*(5303^1)+1 > 1 5 dd **** 2*(5399^1)+1 > 1 5 dd **** 2*(5639^1)+1 > 1 5 dd **** 2*(5711^1)+1 > 1 5 dd **** 2*(5903^1)+1 > 1 5 dd **** 2*(6131^1)+1 > 1 5 dd **** 2*(6263^1)+1 > 1 5 dd **** 2*(6323^1)+1 > 1 5 dd **** 2*(6491^1)+1 > 1 5 dd **** 2*(6551^1)+1 > 1 5 dd **** 2*(6563^1)+1 > 1 5 dd **** 2*(6899^1)+1 > 1 5 dd **** 2*(6983^1)+1 > 1 5 dd **** 2*(7043^1)+1 > 1 5 dd **** 2*(7079^1)+1 > 1 5 dd **** 2*(7103^1)+1 > 1 5 dd **** 2*(7151^1)+1 > 1 5 dd **** 2*(7211^1)+1 > 1 5 dd **** 2*(7643^1)+1 > 1 5 dd **** 2*(7691^1)+1 > 1 5 dd **** 2*(7823^1)+1 > 1 5 dd **** 2*(7883^1)+1 > 1 5 dd **** 2*(8111^1)+1 > 1 5 dd **** 2*(8243^1)+1 > 1 5 dd **** 2*(8663^1)+1 > 1 5 dd **** 2*(8951^1)+1 > 1 5 dd **** 2*(9059^1)+1 > 1 5 dd **** 2*(9371^1)+1 > 1 5 dd **** 2*(9419^1)+1 > 1 5 dd **** 2*(9479^1)+1 > 1 5 dd **** 2*(9539^1)+1 > 1 5 dd **** 2*(9791^1)+1 > 1 5 dd **** 2*(10091^1)+1 > 1 5 dd **** 2*(10163^1)+1 > 1 5 dd **** 2*(10271^1)+1 > 1 5 dd **** 2*(10331^1)+1 > 1 5 dd **** 2*(10691^1)+1 > 1 5 dd **** 2*(10799^1)+1 > 1 5 dd **** 2*(10883^1)+1 > 1 5 dd **** 2*(11171^1)+1 > 1 5 dd **** 2*(11471^1)+1 > 1 5 dd **** 2*(11519^1)+1 > 1 5 dd **** 2*(11579^1)+1 > 1 5 dd **** 2*(11699^1)+1 > 1 5 dd **** 2*(11783^1)+1 > 1 5 dd **** 2*(11831^1)+1 > 1 5 dd **** 2*(11939^1)+1 > 1 5 dd **** 2*(12011^1)+1 > 1 5 dd **** 2*(12119^1)+1 > 1 5 dd **** 2*(12203^1)+1 > 1 5 dd **** 2*(12263^1)+1 > 1 5 dd **** 2*(12671^1)+1 > 1 5 dd **** 2*(12791^1)+1 > 1 5 dd **** 2*(12899^1)+1 > 1 5 dd **** 2*(12923^1)+1 > 1 5 dd **** 2*(12959^1)+1 > 1 5 dd **** 2*(13451^1)+1 > 1 5 dd **** 2*(13463^1)+1 > 1 5 dd **** 2*(13619^1)+1 > 1 5 dd **** 2*(13763^1)+1 > 1 5 dd **** 2*(13883^1)+1 > 1 5 dd **** 2*(14159^1)+1 > 1 5 dd **** 2*(14303^1)+1 > 1 5 dd **** 2*(14699^1)+1 > 1 5 dd **** 2*(14783^1)+1 > 1 5 dd **** 2*(14831^1)+1 > 1 5 dd **** 2*(14879^1)+1 > 1 5 dd **** 2*(14939^1)+1 > 1 5 dd **** 2*(15791^1)+1 > 1 5 dd **** 2*(15803^1)+1 > 1 5 dd **** 2*(15923^1)+1 > 1 5 dd **** 2*(16091^1)+1 > 1 5 dd **** 2*(16811^1)+1 > 1 5 dd **** 2*(16823^1)+1 > 1 5 dd **** 2*(16883^1)+1 > 1 5 dd **** 2*(16931^1)+1 > 1 5 dd **** 2*(17159^1)+1 > 1 5 dd **** 2*(17183^1)+1 > 1 5 dd **** 2*(17291^1)+1 > 1 5 dd **** 2*(17351^1)+1 > 1 5 dd **** 2*(17579^1)+1 > 1 5 dd **** 2*(17939^1)+1 > 1 5 dd **** 2*(18131^1)+1 > 1 5 dd **** 2*(18191^1)+1 > 1 5 dd **** 2*(18443^1)+1 > 1 5 dd **** 2*(18731^1)+1 > 1 5 dd **** 2*(18803^1)+1 > 1 5 dd **** 2*(18899^1)+1 > 1 5 dd **** 2*(19163^1)+1 > 1 5 dd **** 2*(19319^1)+1 > 1 5 dd **** 2*(19391^1)+1 > 1 5 dd **** 2*(19559^1)+1 > 1 5 dd **** 2*(19751^1)+1 > 1 5 dd **** 2*(19919^1)+1 > 1 5 dd **** 2*(19991^1)+1 > 1 5 dd **** 2*(20063^1)+1 > 1 5 dd **** 2*(20411^1)+1 > 1 5 dd **** 2*(20759^1)+1 > 1 5 dd **** 2*(20771^1)+1 > 1 5 dd **** 2*(20879^1)+1 > 1 5 dd **** 2*(20939^1)+1 > 1 5 dd **** 2*(20963^1)+1 > 1 5 dd **** 2*(21011^1)+1 > 1 5 dd **** 2*(21179^1)+1 > 1 5 dd **** 2*(21383^1)+1 > 1 5 dd **** 2*(21419^1)+1 > 1 5 dd **** 2*(21611^1)+1 > 1 5 dd **** 2*(21803^1)+1 > 1 5 dd **** 2*(22079^1)+1 > 1 5 dd **** 2*(22259^1)+1 > 1 5 dd **** 2*(22271^1)+1 > 1 5 dd **** 2*(22343^1)+1 > 1 5 dd **** 2*(22751^1)+1 > 1 5 dd **** 2*(22943^1)+1 > 1 5 dd **** 2*(23099^1)+1 > 1 5 dd **** 2*(23279^1)+1 > 1 5 dd **** 2*(23339^1)+1 > 1 5 dd **** 2*(23459^1)+1 > 1 5 dd **** 2*(23603^1)+1 > 1 5 dd **** 2*(23819^1)+1 > 1 5 dd **** 2*(24203^1)+1 > 1 5 dd **** 2*(24239^1)+1 > 1 5 dd **** 2*(24551^1)+1 > 1 5 dd **** 2*(24611^1)+1 > 1 5 dd **** 2*(24683^1)+1 > 1 5 dd **** 2*(24971^1)+1 > 1 5 dd **** 2*(25523^1)+1 > 1 5 dd **** 2*(25643^1)+1 > 1 5 dd **** 2*(25703^1)+1 > 1 5 dd **** 2*(25799^1)+1 > 1 5 dd **** 2*(25919^1)+1 > 1 5 dd **** 2*(26111^1)+1 > 1 5 dd **** 2*(26459^1)+1 > 1 5 dd **** 2*(26879^1)+1 > 1 5 dd **** 2*(26891^1)+1 > 1 5 dd **** 2*(27143^1)+1 > 1 5 dd **** 2*(27479^1)+1 > 1 5 dd **** 2*(27539^1)+1 > 1 5 dd **** 2*(27551^1)+1 > 1 5 dd **** 2*(27743^1)+1 > 1 5 dd **** 2*(27983^1)+1 > 1 5 dd **** 2*(28019^1)+1 > 1 5 dd **** 2*(28403^1)+1 > 1 5 dd **** 2*(28499^1)+1 > 1 5 dd **** 2*(28559^1)+1 > 1 5 dd **** 2*(28571^1)+1 > 1 5 dd **** 2*(28643^1)+1 > 1 5 dd **** 2*(28751^1)+1 > 1 5 dd **** 2*(28859^1)+1 > 1 5 dd **** 2*(29339^1)+1 > 1 5 dd **** 2*(29363^1)+1 > 1 5 dd **** 2*(29483^1)+1 > 1 5 dd **** 2*(29531^1)+1 > 1 5 dd **** 2*(29723^1)+1 > 1 5 dd **** 2*(30323^1)+1 > 1 5 dd **** 2*(30671^1)+1 > 1 5 dd **** 2*(30851^1)+1 > 1 5 dd **** 2*(30983^1)+1 > 1 5 dd **** 2*(31019^1)+1 > 1 5 dd **** 2*(31151^1)+1 > 1 5 dd **** 2*(31319^1)+1 > 1 5 dd **** 2*(31799^1)+1 > 1 5 dd **** 2*(31859^1)+1 > 1 5 dd **** 2*(32003^1)+1 > 1 5 dd **** 2*(32159^1)+1 > 1 5 dd **** 2*(32531^1)+1 > 1 5 dd **** 2*(32771^1)+1 > 1 5 dd **** 2*(32843^1)+1 > 1 5 dd **** 2*(33023^1)+1 > 1 5 dd **** 2*(33119^1)+1 > 1 5 dd **** 2*(33179^1)+1 > 1 5 dd **** 2*(33191^1)+1 > 1 5 dd **** 2*(33479^1)+1 > 1 5 dd **** 2*(33623^1)+1 > 1 5 dd **** 2*(34283^1)+1 > 1 5 dd **** 2*(34319^1)+1 > 1 5 dd **** 2*(34439^1)+1 > 1 5 dd **** 2*(34631^1)+1 > 1 5 dd **** 2*(34883^1)+1 > 1 5 dd **** 2*(35099^1)+1 > 1 5 dd **** 2*(35111^1)+1 > 1 5 dd **** 2*(35291^1)+1 > 1 5 dd **** 2*(35831^1)+1 > 1 5 dd **** 2*(35999^1)+1 > 1 5 dd **** 2*(36083^1)+1 > 1 5 dd **** 2*(36191^1)+1 > 1 5 dd **** 2*(36251^1)+1 > 1 5 dd **** 2*(36383^1)+1 > 1 5 dd **** 2*(36479^1)+1 > 1 5 dd **** 2*(36563^1)+1 > 1 5 dd **** 2*(36791^1)+1 > 1 5 dd **** 2*(36923^1)+1 > 1 5 dd **** 2*(37139^1)+1 > 1 5 dd **** 2*(37379^1)+1 > 1 5 dd **** 2*(37619^1)+1 > 1 5 dd **** 2*(37871^1)+1 > 1 5 dd **** 2*(37991^1)+1 > 1 5 dd **** 2*(38039^1)+1 > 1 5 dd **** 2*(38183^1)+1 > 1 5 dd **** 2*(38231^1)+1 > 1 5 dd **** 2*(38303^1)+1 > 1 5 dd **** 2*(38459^1)+1 > 1 5 dd **** 2*(38639^1)+1 > 1 5 dd **** 2*(38723^1)+1 > 1 5 dd **** 2*(38891^1)+1 > 1 5 dd **** 2*(39239^1)+1 > 1 5 dd **** 2*(39419^1)+1 > 1 5 dd **** 2*(39443^1)+1 > 1 5 dd **** 2*(39551^1)+1 > 1 5 dd **** 2*(39659^1)+1 > 1 5 dd **** 2*(39779^1)+1 > 1 5 dd **** 2*(39971^1)+1 > 1 5 dd **** 2*(39983^1)+1 > 1 5 dd **** 2*(40283^1)+1 > 1 5 dd **** 2*(40343^1)+1 > 1 5 dd **** 2*(40559^1)+1 > 1 5 dd **** 2*(40763^1)+1 > 1 5 dd **** 2*(40823^1)+1 > 1 5 dd **** 2*(41231^1)+1 > 1 5 dd **** 2*(41243^1)+1 > 1 5 dd **** 2*(41399^1)+1 > 1 5 dd **** 2*(41603^1)+1 > 1 5 dd **** 2*(42023^1)+1 > 1 5 dd **** 2*(42071^1)+1 > 1 5 dd **** 2*(42131^1)+1 > 1 5 dd **** 2*(42359^1)+1 > 1 5 dd **** 2*(42611^1)+1 > 1 5 dd **** 2*(42719^1)+1 > 1 5 dd **** 2*(42743^1)+1 > 1 5 dd **** 2*(42923^1)+1 > 1 5 dd **** 2*(43391^1)+1 > 1 5 dd **** 2*(43691^1)+1 > 1 5 dd **** 2*(43943^1)+1 > 1 5 dd **** 2*(44111^1)+1 > 1 5 dd **** 2*(44543^1)+1 > 1 5 dd **** 2*(44651^1)+1 > 1 5 dd **** 2*(44699^1)+1 > 1 5 dd **** 2*(44879^1)+1 > 1 5 dd **** 2*(45119^1)+1 > 1 5 dd **** 2*(45131^1)+1 > 1 5 dd **** 2*(45179^1)+1 > 1 5 dd **** 2*(45263^1)+1 > 1 5 dd **** 2*(45599^1)+1 > 1 5 dd **** 2*(45971^1)+1 > 1 5 dd **** 2*(46199^1)+1 > 1 5 dd **** 2*(46523^1)+1 > 1 5 dd **** 2*(46619^1)+1 > 1 5 dd **** 2*(46643^1)+1 > 1 5 dd **** 2*(46691^1)+1 > 1 5 dd **** 2*(46703^1)+1 > 1 5 dd **** 2*(46751^1)+1 > 1 5 dd **** 2*(47279^1)+1 > 1 5 dd **** 2*(47363^1)+1 > 1 5 dd **** 2*(47543^1)+1 > 1 5 dd **** 2*(47639^1)+1 > 1 5 dd **** 2*(48131^1)+1 > 1 5 dd **** 2*(48239^1)+1 > 1 5 dd **** 2*(48479^1)+1 > 1 5 dd **** 2*(48563^1)+1 > 1 5 dd **** 2*(48731^1)+1 > 1 5 dd **** 2*(49103^1)+1 > 1 5 dd **** 2*(49331^1)+1 > 1 5 dd **** 2*(49463^1)+1 > 1 5 dd **** 2*(49499^1)+1 > 1 5 dd **** 2*(49559^1)+1 > 1 5 dd **** 2*(49811^1)+1 > 1 5 dd **** 2*(49919^1)+1 > 1 5 dd **** 2*(50051^1)+1 > 1 6 dd **** 2*(50411^1)+1 > 1 6 dd **** 2*(50423^1)+1 > 1 6 dd **** 2*(50591^1)+1 > 1 6 dd **** 2*(51203^1)+1 > 1 6 dd **** 2*(51503^1)+1 > 1 6 dd **** 2*(51539^1)+1 > 1 6 dd **** 2*(51659^1)+1 > 1 6 dd **** 2*(52103^1)+1 > 1 6 dd **** 2*(52163^1)+1 > 1 6 dd **** 2*(52379^1)+1 > 1 6 dd **** 2*(52511^1)+1 > 1 6 dd **** 2*(52571^1)+1 > 1 6 dd **** 2*(52583^1)+1 > 1 6 dd **** 2*(52631^1)+1 > 1 6 dd **** 2*(52883^1)+1 > 1 6 dd **** 2*(53051^1)+1 > 1 6 dd **** 2*(53411^1)+1 > 1 6 dd **** 2*(53591^1)+1 > 1 6 dd **** 2*(53639^1)+1 > 1 6 dd **** 2*(53951^1)+1 > 1 6 dd **** 2*(54011^1)+1 > 1 6 dd **** 2*(54251^1)+1 > 1 6 dd **** 2*(54443^1)+1 > 1 6 dd **** 2*(54959^1)+1 > 1 6 dd **** 2*(55439^1)+1 > 1 6 dd **** 2*(55631^1)+1 > 1 6 dd **** 2*(55799^1)+1 > 1 6 dd **** 2*(55931^1)+1 > 1 6 dd **** 2*(56099^1)+1 > 1 6 dd **** 2*(56123^1)+1 > 1 6 dd **** 2*(56519^1)+1 > 1 6 dd **** 2*(56531^1)+1 > 1 6 dd **** 2*(56663^1)+1 > 1 6 dd **** 2*(56783^1)+1 > 1 6 dd **** 2*(56891^1)+1 > 1 6 dd **** 2*(56951^1)+1 > 1 6 dd **** 2*(57203^1)+1 > 1 6 dd **** 2*(57839^1)+1 > 1 6 dd **** 2*(58211^1)+1 > 1 6 dd **** 2*(58451^1)+1 > 1 6 dd **** 2*(58511^1)+1 > 1 6 dd **** 2*(58979^1)+1 > 1 6 dd **** 2*(59063^1)+1 > 1 6 dd **** 2*(59123^1)+1 > 1 6 dd **** 2*(59399^1)+1 > 1 6 dd **** 2*(59723^1)+1 > 1 6 dd **** 2*(59879^1)+1 > 1 6 dd **** 2*(60083^1)+1 > 1 6 dd **** 2*(60251^1)+1 > 1 6 dd **** 2*(60383^1)+1 > 1 6 dd **** 2*(60719^1)+1 > 1 6 dd **** 2*(60779^1)+1 > 1 6 dd **** 2*(61331^1)+1 > 1 6 dd **** 2*(61703^1)+1 > 1 6 dd **** 2*(61751^1)+1 > 1 6 dd **** 2*(61991^1)+1 > 1 6 dd **** 2*(62099^1)+1 > 1 6 dd **** 2*(62171^1)+1 > 1 6 dd **** 2*(62351^1)+1 > 1 6 dd **** 2*(62423^1)+1 > 1 6 dd **** 2*(62459^1)+1 > 1 6 dd **** 2*(62591^1)+1 > 1 6 dd **** 2*(62603^1)+1 > 1 6 dd **** 2*(62819^1)+1 > 1 6 dd **** 2*(63179^1)+1 > 1 6 dd **** 2*(63419^1)+1 > 1 6 dd **** 2*(63671^1)+1 > 1 6 dd **** 2*(63743^1)+1 > 1 6 dd **** 2*(63803^1)+1 > 1 6 dd **** 2*(63839^1)+1 > 1 6 dd **** 2*(63863^1)+1 > 1 6 dd **** 2*(64439^1)+1 > 1 6 dd **** 2*(64451^1)+1 > 1 6 dd **** 2*(64763^1)+1 > 1 6 dd **** 2*(65063^1)+1 > 1 6 dd **** 2*(65099^1)+1 > 1 6 dd **** 2*(65111^1)+1 > 1 6 dd **** 2*(65171^1)+1 > 1 6 dd **** 2*(65183^1)+1 > 1 6 dd **** 2*(65651^1)+1 > 1 6 dd **** 2*(65843^1)+1 > 1 6 dd **** 2*(65963^1)+1 > 1 6 dd **** 2*(66191^1)+1 > 1 6 dd **** 2*(66431^1)+1 > 1 6 dd **** 2*(66791^1)+1 > 1 6 dd **** 2*(66959^1)+1 > 1 6 dd **** 2*(67043^1)+1 > 1 6 dd **** 2*(67103^1)+1 > 1 6 dd **** 2*(67499^1)+1 > 1 6 dd **** 2*(67559^1)+1 > 1 6 dd **** 2*(67943^1)+1 > 1 6 dd **** 2*(68111^1)+1 > 1 6 dd **** 2*(68171^1)+1 > 1 6 dd **** 2*(68279^1)+1 > 1 6 dd **** 2*(68543^1)+1 > 1 6 dd **** 2*(68639^1)+1 > 1 6 dd **** 2*(68699^1)+1 > 1 6 dd **** 2*(68819^1)+1 > 1 6 dd **** 2*(68963^1)+1 > 1 6 dd **** 2*(69119^1)+1 > 1 6 dd **** 2*(69203^1)+1 > 1 6 dd **** 2*(69431^1)+1 > 1 6 dd **** 2*(69539^1)+1 > 1 6 dd **** 2*(70079^1)+1 > 1 6 dd **** 2*(70379^1)+1 > 1 6 dd **** 2*(70979^1)+1 > 1 6 dd **** 2*(71399^1)+1 > 1 6 dd **** 2*(71843^1)+1 > 1 6 dd **** 2*(71999^1)+1 > 1 6 dd **** 2*(72503^1)+1 > 1 6 dd **** 2*(72911^1)+1 > 1 6 dd **** 2*(73259^1)+1 > 1 6 dd **** 2*(73523^1)+1 > 1 6 dd **** 2*(73751^1)+1 > 1 6 dd **** 2*(73823^1)+1 > 1 6 dd **** 2*(74099^1)+1 > 1 6 dd **** 2*(74219^1)+1 > 1 6 dd **** 2*(74363^1)+1 > 1 6 dd **** 2*(74699^1)+1 > 1 6 dd **** 2*(74759^1)+1 > 1 6 dd **** 2*(74771^1)+1 > 1 6 dd **** 2*(75479^1)+1 > 1 6 dd **** 2*(75503^1)+1 > 1 6 dd **** 2*(75983^1)+1 > 1 6 dd **** 2*(76031^1)+1 > 1 6 dd **** 2*(76091^1)+1 > 1 6 dd **** 2*(76259^1)+1 > 1 6 dd **** 2*(76283^1)+1 > 1 6 dd **** 2*(76679^1)+1 > 1 6 dd **** 2*(76871^1)+1 > 1 6 dd **** 2*(76943^1)+1 > 1 6 dd **** 2*(77243^1)+1 > 1 6 dd **** 2*(77471^1)+1 > 1 6 dd **** 2*(77543^1)+1 > 1 6 dd **** 2*(77699^1)+1 > 1 6 dd **** 2*(77711^1)+1 > 1 6 dd **** 2*(78059^1)+1 > 1 6 dd **** 2*(78311^1)+1 > 1 6 dd **** 2*(78623^1)+1 > 1 6 dd **** 2*(78779^1)+1 > 1 6 dd **** 2*(78839^1)+1 > 1 6 dd **** 2*(79151^1)+1 > 1 6 dd **** 2*(79259^1)+1 > 1 6 dd **** 2*(79283^1)+1 > 1 6 dd **** 2*(79379^1)+1 > 1 6 dd **** 2*(79559^1)+1 > 1 6 dd **** 2*(79811^1)+1 > 1 6 dd **** 2*(80039^1)+1 > 1 6 dd **** 2*(80651^1)+1 > 1 6 dd **** 2*(80819^1)+1 > 1 6 dd **** 2*(81071^1)+1 > 1 6 dd **** 2*(81131^1)+1 > 1 6 dd **** 2*(81563^1)+1 > 1 6 dd **** 2*(81611^1)+1 > 1 6 dd **** 2*(81839^1)+1 > 1 6 dd **** 2*(82139^1)+1 > 1 6 dd **** 2*(82223^1)+1 > 1 6 dd **** 2*(82499^1)+1 > 1 6 dd **** 2*(82763^1)+1 > 1 6 dd **** 2*(83243^1)+1 > 1 6 dd **** 2*(83339^1)+1 > 1 6 dd **** 2*(83399^1)+1 > 1 6 dd **** 2*(83423^1)+1 > 1 6 dd **** 2*(83459^1)+1 > 1 6 dd **** 2*(83579^1)+1 > 1 6 dd **** 2*(83939^1)+1 > 1 6 dd **** 2*(84011^1)+1 > 1 6 dd **** 2*(84131^1)+1 > 1 6 dd **** 2*(84263^1)+1 > 1 6 dd **** 2*(84299^1)+1 > 1 6 dd **** 2*(84431^1)+1 > 1 6 dd **** 2*(84443^1)+1 > 1 6 dd **** 2*(84503^1)+1 > 1 6 dd **** 2*(84659^1)+1 > 1 6 dd **** 2*(85103^1)+1 > 1 6 dd **** 2*(85223^1)+1 > 1 6 dd **** 2*(85523^1)+1 > 1 6 dd **** 2*(85691^1)+1 > 1 6 dd **** 2*(85931^1)+1 > 1 6 dd **** 2*(86111^1)+1 > 1 6 dd **** 2*(86171^1)+1 > 1 6 dd **** 2*(86291^1)+1 > 1 6 dd **** 2*(86771^1)+1 > 1 6 dd **** 2*(86843^1)+1 > 1 6 dd **** 2*(87071^1)+1 > 1 6 dd **** 2*(87299^1)+1 > 1 6 dd **** 2*(87383^1)+1 > 1 6 dd **** 2*(87539^1)+1 > 1 6 dd **** 2*(87959^1)+1 > 1 6 dd **** 2*(88079^1)+1 > 1 6 dd **** 2*(88463^1)+1 > 1 6 dd **** 2*(88811^1)+1 > 1 6 dd **** 2*(88919^1)+1 > 1 6 dd **** 2*(89051^1)+1 > 1 6 dd **** 2*(89123^1)+1 > 1 6 dd **** 2*(89399^1)+1 > 1 6 dd **** 2*(89759^1)+1 > 1 6 dd **** 2*(90011^1)+1 > 1 6 dd **** 2*(90599^1)+1 > 1 6 dd **** 2*(90803^1)+1 > 1 6 dd **** 2*(90971^1)+1 > 1 6 dd **** 2*(91079^1)+1 > 1 6 dd **** 2*(91139^1)+1 > 1 6 dd **** 2*(91463^1)+1 > 1 6 dd **** 2*(91499^1)+1 > 1 6 dd **** 2*(91583^1)+1 > 1 6 dd **** 2*(91631^1)+1 > 1 6 dd **** 2*(91691^1)+1 > 1 6 dd **** 2*(92003^1)+1 > 1 6 dd **** 2*(92243^1)+1 > 1 6 dd **** 2*(92363^1)+1 > 1 6 dd **** 2*(92951^1)+1 > 1 6 dd **** 2*(93059^1)+1 > 1 6 dd **** 2*(93239^1)+1 > 1 6 dd **** 2*(93323^1)+1 > 1 6 dd **** 2*(93371^1)+1 > 1 6 dd **** 2*(93479^1)+1 > 1 6 dd **** 2*(93563^1)+1 > 1 6 dd **** 2*(93683^1)+1 > 1 6 dd **** 2*(93911^1)+1 > 1 6 dd **** 2*(94079^1)+1 > 1 6 dd **** 2*(94151^1)+1 > 1 6 dd **** 2*(94343^1)+1 > 1 6 dd **** 2*(94463^1)+1 > 1 6 dd **** 2*(95723^1)+1 > 1 6 dd **** 2*(95891^1)+1 > 1 6 dd **** 2*(96443^1)+1 > 1 6 dd **** 2*(96731^1)+1 > 1 6 dd **** 2*(96779^1)+1 > 1 6 dd **** 2*(96851^1)+1 > 1 6 dd **** 2*(97511^1)+1 > 1 6 dd **** 2*(97523^1)+1 > 1 6 dd **** 2*(97871^1)+1 > 1 6 dd **** 2*(97931^1)+1 > 1 6 dd **** 2*(97943^1)+1 > 1 6 dd **** 2*(98123^1)+1 > 1 6 dd **** 2*(98459^1)+1 > 1 6 dd **** 2*(98639^1)+1 > 1 6 dd **** 2*(98711^1)+1 > 1 6 dd **** 2*(98963^1)+1 > 1 6 dd **** 2*(99023^1)+1 > 1 6 dd **** 2*(99251^1)+1 > 1 6 dd **** 2*(99551^1)+1 > 1 6 dd **** 2*(99623^1)+1 > 1 6 dd **** 2*(99839^1)+1 > 1 6 dd **** 2*(100043^1)+1 > 1 6 dd **** 2*(100403^1)+1 > 1 6 dd **** 2*(100559^1)+1 > 1 6 dd **** 2*(100799^1)+1 > 1 6 dd **** 2*(100811^1)+1 > 1 6 dd **** 2*(101063^1)+1 > 1 6 dd **** 2*(101399^1)+1 > 1 6 dd **** 2*(101411^1)+1 > 1 6 dd **** 2*(101483^1)+1 > 1 6 dd **** 2*(101603^1)+1 > 1 6 dd **** 2*(101999^1)+1 > 1 6 dd **** 2*(102023^1)+1 > 1 6 dd **** 2*(102071^1)+1 > 1 6 dd **** 2*(102299^1)+1 > 1 6 dd **** 2*(102359^1)+1 > 1 6 dd **** 2*(102551^1)+1 > 1 6 dd **** 2*(102611^1)+1 > 1 6 dd **** 2*(102911^1)+1 > 1 6 dd **** 2*(102983^1)+1 > 1 6 dd **** 2*(103091^1)+1 > 1 6 dd **** 2*(103319^1)+1 > 1 6 dd **** 2*(103391^1)+1 > 1 6 dd **** 2*(103619^1)+1 > 1 6 dd **** 2*(103643^1)+1 > 1 6 dd **** 2*(104183^1)+1 > 1 6 dd **** 2*(104231^1)+1 > 1 6 dd **** 2*(104399^1)+1 > 1 6 dd **** 2*(104579^1)+1 > 1 6 dd **** 2*(104759^1)+1 > 1 6 dd **** 2*(105071^1)+1 > 1 6 dd **** 2*(105263^1)+1 > 1 6 dd **** 2*(105359^1)+1 > 1 6 dd **** 2*(105503^1)+1 > 1 6 dd **** 2*(105863^1)+1 > 1 6 dd **** 2*(105971^1)+1 > 1 6 dd **** 2*(106019^1)+1 > 1 6 dd **** 2*(106103^1)+1 > 1 6 dd **** 2*(106451^1)+1 > 1 6 dd **** 2*(106703^1)+1 > 1 6 dd **** 2*(107279^1)+1 > 1 6 dd **** 2*(107699^1)+1 > 1 6 dd **** 2*(107843^1)+1 > 1 6 dd **** 2*(108011^1)+1 > 1 6 dd **** 2*(108131^1)+1 > 1 6 dd **** 2*(108359^1)+1 > 1 6 dd **** 2*(108863^1)+1 > 1 6 dd **** 2*(109139^1)+1 > 1 6 dd **** 2*(109211^1)+1 > 1 6 dd **** 2*(109391^1)+1 > 1 6 dd **** 2*(109751^1)+1 > 1 6 dd **** 2*(109883^1)+1 > 1 6 dd **** 2*(109919^1)+1 > 1 6 dd **** 2*(110459^1)+1 > 1 6 dd **** 2*(110543^1)+1 > 1 6 dd **** 2*(110651^1)+1 > 1 6 dd **** 2*(111263^1)+1 > 1 6 dd **** 2*(111323^1)+1 > 1 6 dd **** 2*(111431^1)+1 > 1 6 dd **** 2*(111623^1)+1 > 1 6 dd **** 2*(111659^1)+1 > 1 6 dd **** 2*(111731^1)+1 > 1 6 dd **** 2*(111959^1)+1 > 1 6 dd **** 2*(112163^1)+1 > 1 6 dd **** 2*(112559^1)+1 > 1 6 dd **** 2*(112571^1)+1 > 1 6 dd **** 2*(112583^1)+1 > 1 6 dd **** 2*(112643^1)+1 > 1 6 dd **** 2*(112691^1)+1 > 1 6 dd **** 2*(112919^1)+1 > 1 6 dd **** 2*(113051^1)+1 > 1 6 dd **** 2*(113759^1)+1 > 1 6 dd **** 2*(113783^1)+1 > 1 6 dd **** 2*(114479^1)+1 > 1 6 dd **** 2*(114599^1)+1 > 1 6 dd **** 2*(114671^1)+1 > 1 6 dd **** 2*(114743^1)+1 > 1 6 dd **** 2*(115151^1)+1 > 1 6 dd **** 2*(115163^1)+1 > 1 6 dd **** 2*(115331^1)+1 > 1 6 dd **** 2*(115499^1)+1 > 1 6 dd **** 2*(115679^1)+1 > 1 6 dd **** 2*(115751^1)+1 > 1 6 dd **** 2*(115859^1)+1 > 1 6 dd **** 2*(116243^1)+1 > 1 6 dd **** 2*(116411^1)+1 > 1 6 dd **** 2*(116423^1)+1 > 1 6 dd **** 2*(116579^1)+1 > 1 6 dd **** 2*(116639^1)+1 > 1 6 dd **** 2*(116663^1)+1 > 1 6 dd **** 2*(117119^1)+1 > 1 6 dd **** 2*(117191^1)+1 > 1 6 dd **** 2*(117371^1)+1 > 1 6 dd **** 2*(117431^1)+1 > 1 6 dd **** 2*(117503^1)+1 > 1 6 dd **** 2*(117779^1)+1 > 1 6 dd **** 2*(117839^1)+1 > 1 6 dd **** 2*(117959^1)+1 > 1 6 dd **** 2*(118043^1)+1 > 1 6 dd **** 2*(118259^1)+1 > 1 6 dd **** 2*(118571^1)+1 > 1 6 dd **** 2*(119039^1)+1 > 1 6 dd **** 2*(119363^1)+1 > 1 6 dd **** 2*(119771^1)+1 > 1 6 dd **** 2*(119783^1)+1 > 1 6 dd **** 2*(119891^1)+1 > 1 6 dd **** 2*(120299^1)+1 > 1 6 dd **** 2*(120371^1)+1 > 1 6 dd **** 2*(120539^1)+1 > 1 6 dd **** 2*(120563^1)+1 > 1 6 dd **** 2*(120671^1)+1 > 1 6 dd **** 2*(120779^1)+1 > 1 6 dd **** 2*(120863^1)+1 > 1 6 dd **** 2*(121139^1)+1 > 1 6 dd **** 2*(121259^1)+1 > 1 6 dd **** 2*(121403^1)+1 > 1 6 dd **** 2*(121559^1)+1 > 1 6 dd **** 2*(121631^1)+1 > 1 6 dd **** 2*(121763^1)+1 > 1 6 dd **** 2*(121931^1)+1 > 1 6 dd **** 2*(122099^1)+1 > 1 6 dd **** 2*(122231^1)+1 > 1 6 dd **** 2*(122471^1)+1 > 1 6 dd **** 2*(122819^1)+1 > 1 6 dd **** 2*(122891^1)+1 > 1 6 dd **** 2*(123059^1)+1 > 1 6 dd **** 2*(123083^1)+1 > 1 6 dd **** 2*(123419^1)+1 > 1 6 dd **** 2*(123503^1)+1 > 1 6 dd **** 2*(123719^1)+1 > 1 6 dd **** 2*(123731^1)+1 > 1 6 dd **** 2*(123803^1)+1 > 1 6 dd **** 2*(123923^1)+1 > 1 6 dd **** 2*(124643^1)+1 > 1 6 dd **** 2*(124823^1)+1 > 1 6 dd **** 2*(125003^1)+1 > 1 6 dd **** 2*(125399^1)+1 > 1 6 dd **** 2*(126443^1)+1 > 1 6 dd **** 2*(126491^1)+1 > 1 6 dd **** 2*(126551^1)+1 > 1 6 dd **** 2*(126683^1)+1 > 1 6 dd **** 2*(126719^1)+1 > 1 6 dd **** 2*(126839^1)+1 > 1 6 dd **** 2*(126851^1)+1 > 1 6 dd **** 2*(127103^1)+1 > 1 6 dd **** 2*(127139^1)+1 > 1 6 dd **** 2*(127331^1)+1 > 1 6 dd **** 2*(127691^1)+1 > 1 6 dd **** 2*(128099^1)+1 > 1 6 dd **** 2*(128399^1)+1 > 1 6 dd **** 2*(128483^1)+1 > 1 6 dd **** 2*(128819^1)+1 > 1 6 dd **** 2*(128939^1)+1 > 1 6 dd **** 2*(128951^1)+1 > 1 6 dd **** 2*(129011^1)+1 > 1 6 dd **** 2*(129263^1)+1 > 1 6 dd **** 2*(129443^1)+1 > 1 6 dd **** 2*(129491^1)+1 > 1 6 dd **** 2*(129971^1)+1 > 1 6 dd **** 2*(130199^1)+1 > 1 6 dd (16:50) gp > ## *** last result computed in 172 ms. (16:50) gp > isprime(thePrime ) %20 = 1 (16:50) gp > ## *** last result computed in 0 ms. 6 dd CBZ-110-A Primality test by Rashid Naimi Order = 1 k = 2 [/CODE] |
Pari-GP code:
[CODE] print("CBZ-120-A Primality test by Rashid Naimi") theOrder = 1 \\\\\ k= 22 \\ theLimit =3 \\\ thePrime = 1; forprime(q=theLimit,theLimit+19^4,{ p=k*(q^theOrder )+1; n = Mod(2,p)^(q^theOrder )-1; if(gcd(lift(n),p)>1, print("\n**** ", k,"*(",q,"^",theOrder,")+1 > ",primeFlag); thePrime =p; primeFlag=isprime(p); print(#digits(thePrime)," dd"); \\next(19); \\ Disactivate to scan the entire range ); }) ## isprime(thePrime ) ## print(#digits(thePrime)," dd") print("CBZ-120-A Primality test by Rashid Naimi") print("Order = ",theOrder) print("k = ",k) [/CODE] [CODE] CBZ-120-A Primality test by Rashid Naimi **** 22*(1789^1)+1 > primeFlag 5 dd **** 22*(4153^1)+1 > 1 5 dd **** 22*(6373^1)+1 > 1 6 dd **** 22*(8761^1)+1 > 1 6 dd **** 22*(8929^1)+1 > 1 6 dd **** 22*(10993^1)+1 > 1 6 dd **** 22*(12049^1)+1 > 1 6 dd **** 22*(13669^1)+1 > 1 6 dd **** 22*(14149^1)+1 > 1 6 dd **** 22*(16573^1)+1 > 1 6 dd **** 22*(20809^1)+1 > 1 6 dd **** 22*(22453^1)+1 > 1 6 dd **** 22*(25621^1)+1 > 1 6 dd **** 22*(25933^1)+1 > 1 6 dd **** 22*(26041^1)+1 > 1 6 dd **** 22*(27109^1)+1 > 1 6 dd **** 22*(31321^1)+1 > 1 6 dd **** 22*(34141^1)+1 > 1 6 dd **** 22*(35449^1)+1 > 1 6 dd **** 22*(36061^1)+1 > 1 6 dd **** 22*(36229^1)+1 > 1 6 dd **** 22*(38449^1)+1 > 1 6 dd **** 22*(38629^1)+1 > 1 6 dd **** 22*(39901^1)+1 > 1 6 dd **** 22*(42409^1)+1 > 1 6 dd **** 22*(45841^1)+1 > 1 7 dd **** 22*(46933^1)+1 > 1 7 dd **** 22*(47569^1)+1 > 1 7 dd **** 22*(51769^1)+1 > 1 7 dd **** 22*(52981^1)+1 > 1 7 dd **** 22*(55813^1)+1 > 1 7 dd **** 22*(57373^1)+1 > 1 7 dd **** 22*(57781^1)+1 > 1 7 dd **** 22*(58189^1)+1 > 1 7 dd **** 22*(58573^1)+1 > 1 7 dd **** 22*(60133^1)+1 > 1 7 dd **** 22*(60373^1)+1 > 1 7 dd **** 22*(60589^1)+1 > 1 7 dd **** 22*(60793^1)+1 > 1 7 dd **** 22*(63493^1)+1 > 1 7 dd **** 22*(67429^1)+1 > 1 7 dd **** 22*(70549^1)+1 > 1 7 dd **** 22*(74161^1)+1 > 1 7 dd **** 22*(75013^1)+1 > 1 7 dd **** 22*(77521^1)+1 > 1 7 dd **** 22*(77929^1)+1 > 1 7 dd **** 22*(78781^1)+1 > 1 7 dd **** 22*(80329^1)+1 > 1 7 dd **** 22*(82009^1)+1 > 1 7 dd **** 22*(82129^1)+1 > 1 7 dd **** 22*(82609^1)+1 > 1 7 dd **** 22*(83221^1)+1 > 1 7 dd **** 22*(90469^1)+1 > 1 7 dd **** 22*(91141^1)+1 > 1 7 dd **** 22*(94153^1)+1 > 1 7 dd **** 22*(94693^1)+1 > 1 7 dd **** 22*(94993^1)+1 > 1 7 dd **** 22*(95569^1)+1 > 1 7 dd **** 22*(95581^1)+1 > 1 7 dd **** 22*(95629^1)+1 > 1 7 dd **** 22*(99241^1)+1 > 1 7 dd **** 22*(104869^1)+1 > 1 7 dd **** 22*(108109^1)+1 > 1 7 dd **** 22*(115153^1)+1 > 1 7 dd **** 22*(118249^1)+1 > 1 7 dd **** 22*(120661^1)+1 > 1 7 dd **** 22*(123049^1)+1 > 1 7 dd (16:54) gp > ## *** last result computed in 0 ms. (16:54) gp > isprime(thePrime ) %7 = 1 (16:54) gp > ## *** last result computed in 0 ms. (16:54) gp > print(#digits(thePrime)," dd") 7 dd (16:54) gp > print("CBZ-120-A Primality test by Rashid Naimi") CBZ-120-A Primality test by Rashid Naimi (16:54) gp > print("Order = ",theOrder) Order = 1 k = 22 [/CODE] |
Pari-GP code:
[CODE] print("CBZ-130-A Primality test by Rashid Naimi") theOrder = 157 \\\\\ k= 10 \\ theLimit =3 \\\ thePrime = 1; forprime(q=theLimit,theLimit+19^4,{ p=k*(q^theOrder )+1; n = Mod(2,p)^(q^theOrder )-1; if(gcd(lift(n),p)>1, print("\n**** ", k,"*(",q,"^",theOrder,")+1 > ",primeFlag); thePrime =p; primeFlag=isprime(p); print(#digits(thePrime)," dd"); \\next(19); \\ Disactivate to scan the entire range ); }) ## isprime(thePrime ) ## print(#digits(thePrime)," dd") print("CBZ-130-A Primality test by Rashid Naimi") print("Order = ",theOrder) print("k = ",k) [/CODE]Note: The results below are generated by a slightly different code than above. However the code above can generate the same results with the forprime () bounds adjusted. [CODE] **** 10*(15667^157)+1 660 dd **** 10*(213727^157)+1 838 dd **** 10*(249499^157)+1 849 dd **** 10*(290359^157)+1 859 dd **** 10*(385039^157)+1 878 dd **** 10*(406423^157)+1 882 dd **** 10*(460099^157)+1 891 dd **** 10*(470347^157)+1 892 dd **** 10*(852559^157)+1 933 dd **** 10*(1264063^157)+1 959 dd **** 10*(1442887^157)+1 968 dd **** 10*(1967479^157)+1 990 dd **** 10*(2136439^157)+1 995 dd **** 10*(2180923^157)+1 997 dd **** 10*(2375743^157)+1 1003 dd **** 10*(3111931^157)+1 1021 dd **** 10*(3522583^157)+1 1029 dd **** 10*(3914083^157)+1 1037 dd **** 10*(4042063^157)+1 1039 dd **** 10*(4185751^157)+1 1041 dd **** 10*(4218007^157)+1 1042 dd **** 10*(4522411^157)+1 1046 dd **** 10*(4688683^157)+1 1049 dd **** 10*(4940983^157)+1 1052 dd **** 10*(4970587^157)+1 1053 dd **** 10*(5233411^157)+1 1056 dd **** 10*(5416039^157)+1 1059 dd **** 10*(5547007^157)+1 1060 dd **** 10*(5571763^157)+1 1061 dd **** 10*(5622583^157)+1 1061 dd **** 10*(6039871^157)+1 1066 dd **** 10*(6156919^157)+1 1067 dd **** 10*(6178639^157)+1 1068 dd **** 10*(6255859^157)+1 1069 dd **** 10*(6335599^157)+1 1069 dd **** 10*(6362623^157)+1 1070 dd **** 10*(6535831^157)+1 1072 dd **** 10*(6790363^157)+1 1074 dd **** 10*(6916411^157)+1 1075 dd **** 10*(7020103^157)+1 1076 dd **** 10*(7142623^157)+1 1078 dd **** 10*(7367959^157)+1 1080 dd **** 10*(7461031^157)+1 1081 dd **** 10*(7673011^157)+1 1082 dd **** 10*(7759483^157)+1 1083 dd **** 10*(8143351^157)+1 1086 dd **** 10*(8166451^157)+1 1087 dd **** 10*(8211871^157)+1 1087 dd **** 10*(8348299^157)+1 1088 dd **** 10*(8666839^157)+1 1091 dd **** 10*(8930839^157)+1 1093 dd **** 10*(8946799^157)+1 1093 dd **** 10*(9018727^157)+1 1093 dd **** 10*(9116203^157)+1 1094 dd **** 10*(9326203^157)+1 1096 dd **** 10*(9457963^157)+1 1097 dd **** 10*(9562999^157)+1 1097 dd **** 10*(10158667^157)+1 1102 dd **** 10*(10210351^157)+1 1102 dd **** 10*(10345879^157)+1 1103 dd **** 10*(10488139^157)+1 1104 dd **** 10*(10532239^157)+1 1104 dd **** 10*(10805959^157)+1 1106 dd **** 10*(10814203^157)+1 1106 dd **** 10*(10910743^157)+1 1106 dd **** 10*(10955251^157)+1 1107 dd **** 10*(11675383^157)+1 1111 dd **** 10*(11785303^157)+1 1112 dd **** 10*(11859247^157)+1 1112 dd **** 10*(12191563^157)+1 1114 dd **** 10*(12554827^157)+1 1116 dd **** 10*(12585667^157)+1 1116 dd **** 10*(12884371^157)+1 1118 dd **** 10*(13025899^157)+1 1119 dd **** 10*(13362067^157)+1 1120 dd [/CODE] Largest prime of the form I have found so far: 8*(2213314919151113^127)+1 1950 dd |
I just noticed a non consequential error in the above codes:
the line: primeFlag=isprime(p); will have to be moved up 2 lines as the line above the print(), otherwise the print command will print the primality-truth of the previous prime.:smile: the corrected general code: [CODE] print("CBZ-130-B Primality test by Rashid Naimi") theOrder = 157 \\\\\ k= 10 \\ theLimit =3 \\\ thePrime = 1; forprime(q=theLimit,theLimit+19^4,{ p=k*(q^theOrder )+1; n = Mod(2,p)^(q^theOrder )-1; if(gcd(lift(n),p)>1, primeFlag=isprime(p); print("\n**** ", k,"*(",q,"^",theOrder,")+1 > ",primeFlag); thePrime =p; print(#digits(thePrime)," dd"); \\next(19); \\ Disactivate to scan the entire range ); }) ## isprime(thePrime ) ## print(#digits(thePrime)," dd") print("CBZ-130-B Primality test by Rashid Naimi") print("Order = ",theOrder) print("k = ",k) [/CODE] |
[QUOTE=a1call;486524]I just noticed a non consequential error in the above codes:
the line: primeFlag=isprime(p); will have to be moved up 2 lines as the line above the print(), otherwise the print command will print the primality-truth of the previous prime.:smile: the corrected general code: [CODE] print("CBZ-130-B Primality test by Rashid Naimi") theOrder = 157 \\\\\ k= 10 \\ theLimit =3 \\\ thePrime = 1; forprime(q=theLimit,theLimit+19^4,{ p=k*(q^theOrder )+1; n = Mod(2,p)^(q^theOrder )-1; if(gcd(lift(n),p)>1, primeFlag=isprime(p); print("\n**** ", k,"*(",q,"^",theOrder,")+1 > ",primeFlag); thePrime =p; print(#digits(thePrime)," dd"); \\next(19); \\ Disactivate to scan the entire range ); }) ## isprime(thePrime ) ## print(#digits(thePrime)," dd") print("CBZ-130-B Primality test by Rashid Naimi") print("Order = ",theOrder) print("k = ",k) [/CODE][/QUOTE] Can you post a link to the python code of this one. I do wish to go faster if possible. |
Python is not one of the languages I know.:smile:
|
[QUOTE=a1call;486524]the corrected general code:
[CODE] print("CBZ-130-B Primality test by Rashid Naimi") theOrder = 157 \\\\\ k= 10 \\ theLimit =3 \\\ thePrime = 1; forprime(q=theLimit,theLimit+19^4,{ p=k*(q^theOrder )+1; n = Mod(2,p)^(q^theOrder )-1; if(gcd(lift(n),p)>1, primeFlag=isprime(p); print("\n**** ", k,"*(",q,"^",theOrder,")+1 > ",primeFlag); thePrime =p; print(#digits(thePrime)," dd"); \\next(19); \\ Disactivate to scan the entire range ); }) ## isprime(thePrime ) ## print(#digits(thePrime)," dd") print("CBZ-130-B Primality test by Rashid Naimi") print("Order = ",theOrder) print("k = ",k) [/CODE][/QUOTE] It's a little hard for me to understand you here, since your code doesn't have clearly defined input or output. Let me see if I've properly understood. If k, theOrder, and q are positive integers and [CODE]p = k*(q^theOrder)+1; n = Mod(2,p)^(q^theOrder)-1;[/CODE] then p is prime if and only if [CODE]gcd(lift(n),p)>1[/CODE] Is this correct? In other words, [code]NaimiB(k,q,o)=my(Q=q^o, p=k*Q+1, n=Mod(2,p)^Q-1, g=gcd(lift(n),p)); g>1[/code] returns 1 if k*q^o+1 is prime and 0 otherwise, assuming k, q, and o are positive integers. I want to make sure I understand your claim. Are there other conditions? Does q need to be prime (or odd)? Are there any other conditions on the variables? |
As per SM's link:
[URL]http://primes.utm.edu/notes/proofs/MerDiv2.html[/URL] 23 is prime because it is equal to 2*11+1 and it divides 2^11-1 The proof is by Lagrange. The concept works for [U]certain other values[/U][U] (such as 22 in post number 2)[/U] other than 2 in 2p+1. I have no proof for those but the code outputs [U]only primes[/U], as long as I have ran it. While there is no proof for other values than 2, it is easy to prove each output is prime since all the prime factors of the candidate-1 are known. Additionally the concept works for higher orders than 1 as well. This makes the concept useful for finding primes of the form k*p^n+1 which can be significantly larger than the known primes p. for example: 6*(247607^200)+1 is a 1080 decimal digits prime number obtained from the 6 dd prime 247607. It is a prime number because it divides 2^(247607^200)-1 and k=6 is a multiplier that will always yield a prime number of the form 6p^200+1 if 6p^200+1 | 2^(p^200)-1 Not all values of k will work for any given value of theOrder Here theOrder is 200 and k =6 Additionally the code uses Mod(2,candidate)^theOrder as a powerMod operation to avoid calculation pf unnecessarily large 2^(p^theOrder)-1 I hope that clears things up.:smile: |
[QUOTE=a1call;486551]As per SM's link:
[URL]http://primes.utm.edu/notes/proofs/MerDiv2.html[/URL] 23 is prime because it is equal to 2*11+1 and it divides 2^11-1 The proof is by Lagrange. The concept works for [U]certain other values[/U][U] (such as 22 in post number 2)[/U] other than 2 in 2p+1. [snip][/QUOTE] Depends what you mean by "the concept." If q is prime, and q == 3 (mod 4), then p = 2*q + 1 is prime if and only if p divides 2^q - 1. With p = k*q + 1 for larger k, it is true (assuming q is large enough, say q > k) that p divides 2^q - 1 only if p is prime but it may well happen that q and p = k*q + 1 are both prime, but p does [i]not[/i] divide 2^q - 1. For example, q = 31 and p = 22*31 + 1 = 683 are both prime, but 683 does [i]not[/i] divide 2^31 - 1 which Euler proved to be prime. You can insure that, if p = 22*q + 1 is prime, 2 is a [i]quadratic[/i] residue (mod p) if q == 7 (mod 8). However, insuring that 2 is also an 11th power residue takes us out of the realm of rational integer congruences... |
For the correct theOrder-k pairs, if the test result is positive then kp+1 is definitely prime. However if for the same the result is negative then the p may or may not be composite.
In other words, if (note only one f here) 22*p+1 | 2^p-1 Then 22*p+1 is definitely prime. I do not have a proof for this since 22 is not 2. However I posit that you do not have a counter-example either. Please see post number 2 and the resulting output. The same is true for other theOrder-k pairs such as 157,10 the result of which is shown on post number 3. To repeat for clarity, for a/any given theOrder value there are k values which will work all the time and there are other k values which will not. k=2 and k=22 will work for theOrder=1 all the time. Similarly, k=10 will work for theOrder=157 all the time. Please see post number 3. |
[QUOTE=a1call;486584]For the correct theOrder-k pairs, if the test result is positive then kp+1 is definitely prime. However if for the same the result is negative then the p may or may not be composite.
In other words, if (note only one f here) 22*p+1 | 2^p-1 Then 22*p+1 is definitely prime. I do not have a proof for this since 22 is not 2. However I posit that you do not have a counter-example either. Please see post number 2 and the resulting output. The same is true for other theOrder-k pairs such as 157,10 the result of which is shown on post number 3. To repeat for clarity, for a/any given theOrder value there are k values which will work all the time and there are other k values which will not. k=2 and k=22 will work for theOrder=1 all the time. Similarly, k=10 will work for theOrder=157 all the time. Please see post number 3.[/QUOTE] Please see [URL="http://www.mersenneforum.org/showthread.php?t=17126"]Possible values for k (in Mersenne factors)[/URL] |
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