? znprimroot(101) %1 = Mod(2, 101) ? g=% %2 = Mod(2, 101) ? g^37 %3 = Mod(55, 101) ? znlog(Mod(55,101),g) %4 = 37 ? p=nextprime(10^9) %5 = 1000000007 ? g=znprimroot(p) %6 = Mod(5, 1000000007) ? znlog(Mod(3,p),g) %7 = 884237698 ? p=nextprime(10^99) %8 = 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000289 ? g=znprimroot(p) *** znprimroot: user interrupt after 31,853 ms. ? g=Mod(2,p) %9 = Mod(2, 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000289) ? znlog(Mod(3,p),g) *** znlog: user interrupt after 19,638 ms. ? factor(p-1) *** factor: user interrupt after 12,236 ms. ? p=nextprime(p+1) %10 = 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000303 ? g=Mod(2,p) %11 = Mod(2, 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000303) ? znlog(Mod(3,p),g) *** znlog: user interrupt after 5,297 ms. ? e=ellinit([0,0,0,Mod(3,101),Mod(5,101)]) %12 = [0, 0, 0, Mod(3, 101), Mod(5, 101), 0, Mod(6, 101), Mod(20, 101), Mod(92, 101), Mod(58, 101), Mod(23, 101), Mod(97, 101), Mod(5, 101), 0, 0, 0, 0, 0, 0] ? for(n=1,10,if(issquare(Mod(n^3+3*n+5,101)),print(n))) 1 2 4 6 8 9 10 ? sqrt(Mod(1+3+5,101)) %13 = Mod(3, 101) ? P=[Mod(1,101),Mod(3,101)] %14 = [Mod(1, 101), Mod(3, 101)] ? ellisoncurve(e,P) %15 = 1 ? ellisoncurve(e,[Mod(0,101),Mod(3,101)]) %16 = 0 ? ellpow(e,P,2) %17 = [Mod(100, 101), Mod(100, 101)] ? ellpow(e,P,3) %18 = [Mod(4, 101), Mod(92, 101)] ? ellpow(e,P,65) %19 = [Mod(11, 101), Mod(64, 101)] ? ellap(e,101) %20 = -13 ? ellpow(e,P,115) %21 = [0] ? factor(115) %22 = [5 1] [23 1] ? ellpow(e,P,5) %23 = [Mod(44, 101), Mod(73, 101)] ? ellpow(e,P,23) %24 = [0] ? sqrt(Mod(2^3+3*2+5,101)) %25 = Mod(25, 101) ? Q=[Mod(2,101),%] %26 = [Mod(2, 101), Mod(25, 101)] ? ellisoncurve(Q)  *** expected character: ',' instead of: ellisoncurve(Q) ^- ? ellisoncurve(e,Q) %27 = 1 ? ellpow(e,Q,23) %28 = [Mod(70, 101), Mod(44, 101)] ? ellpow(e,Q,5) %29 = [Mod(62, 101), Mod(19, 101)] ? P %30 = [Mod(1, 101), Mod(3, 101)] ? elladd(e,P,Q) %31 = [Mod(77, 101), Mod(42, 101)] ? ? Help topics: for a list of relevant subtopics, type ?n for n in 0: user-defined identifiers (variable, alias, function) 1: Standard monadic or dyadic OPERATORS 2: CONVERSIONS and similar elementary functions 3: TRANSCENDENTAL functions 4: NUMBER THEORETICAL functions 5: Functions related to ELLIPTIC CURVES 6: Functions related to general NUMBER FIELDS 7: POLYNOMIALS and power series 8: Vectors, matrices, LINEAR ALGEBRA and sets 9: SUMS, products, integrals and similar functions 10: GRAPHIC functions 11: PROGRAMMING under GP 12: The PARI community Also: ? functionname (short on-line help) ?\ (keyboard shortcuts) ?. (member functions) Extended help looks available: ?? (opens the full user's manual in a dvi previewer) ?? tutorial / refcard / libpari (tutorial/reference card/libpari manual) ?? keyword (long help text about "keyword" from the user's manual) ??? keyword (a propos: list of related functions). ? ?5 elladd ellak ellan ellap ellbil ellchangecurve ellchangepoint ellconvertname elleisnum elleta ellgenerators ellglobalred ellheight ellheightmatrix ellidentify ellinit ellisoncurve ellj elllocalred elllseries ellminimalmodel ellorder ellordinate ellpointtoz ellpow ellrootno ellsearch ellsigma ellsub elltaniyama elltors ellwp ellzeta ellztopoint ? ?elladd elladd(e,z1,z2): sum of the points z1 and z2 on elliptic curve e. ? for(n=1,115,if(ellpow(e,Q,n)==P,print(n))) 100 ? for(n=1,115,if(ellpow(e,Q,n)==P,return(n))) %32 = 100 ? r=0;for(n=1,115,if(ellpow(e,Q,n)==P,r=n));r %33 = 100 ? r %34 = 100 ? z=Mod(Mod(1,101)*x,Mod(1,101)*(x^2-t^3-3*t-5)) %35 = Mod(Mod(1, 101)*x, Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101))) ? R=[Mod(Mod(1,101)*t,Mod(1,101)*(x^2-t^3-3*t-5)),z] %36 = [Mod(Mod(1, 101)*t, Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101))), Mod(Mod(1, 101)*x, Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101)))] ? ellisoncurve(e,R) %37 = 1 ? ellpow(e,R,2) %38 = [Mod((Mod(4, 101)*t^4 + Mod(77, 101)*t^2 + Mod(42, 101)*t + Mod(36, 101))/(Mod(16, 101)*t^3 + Mod(48, 101)*t + Mod(80, 101)), Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101))), Mod(((Mod(93, 101)*t^6 + Mod(82, 101)*t^4 + Mod(8, 101)*t^3 + Mod(57, 101)*t^2 + Mod(76, 101)*t + Mod(99, 101))/(Mod(37, 101)*t^6 + Mod(20, 101)*t^4 + Mod(67, 101)*t^3 + Mod(30, 101)*t^2 + Mod(100, 101)*t + Mod(16, 101)))*x, Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101)))] ? lift(%38[1]) %39 = (Mod(4, 101)*t^4 + Mod(77, 101)*t^2 + Mod(42, 101)*t + Mod(36, 101))/(Mod(16, 101)*t^3 + Mod(48, 101)*t + Mod(80, 101)) ? lift(%) %40 = (4*t^4 + 77*t^2 + 42*t + 36)/(16*t^3 + 48*t + 80) ? ellpow(e,R,3) %41 = [Mod((Mod(22, 101)*t^9 + Mod(16, 101)*t^7 + Mod(45, 101)*t^6 + Mod(82, 101)*t^5 + Mod(59, 101)*t^4 + Mod(11, 101)*t^3 + Mod(50, 101)*t^2 + Mod(11, 101)*t + Mod(83, 101))/(Mod(97, 101)*t^8 + Mod(53, 101)*t^6 + Mod(42, 101)*t^5 + Mod(82, 101)*t^4 + Mod(50, 101)*t^3 + Mod(59, 101)*t^2 + Mod(76, 101)*t + Mod(65, 101)), Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101))), Mod(((Mod(89, 101)*t^12 + Mod(16, 101)*t^10 + Mod(31, 101)*t^9 + Mod(44, 101)*t^8 + Mod(100, 101)*t^7 + Mod(38, 101)*t^6 + Mod(52, 101)*t^5 + Mod(86, 101)*t^4 + Mod(65, 101)*t^3 + Mod(61, 101)*t^2 + Mod(48, 101)*t + Mod(66, 101))/(Mod(80, 101)*t^12 + Mod(26, 101)*t^10 + Mod(53, 101)*t^9 + Mod(42, 101)*t^8 + Mod(30, 101)*t^7 + Mod(4, 101)*t^6 + Mod(75, 101)*t^5 + Mod(73, 101)*t^4 + Mod(75, 101)*t^3 + Mod(84, 101)*t^2 + Mod(73, 101)*t + Mod(62, 101)))*x, Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101)))] ? lift(lift(%[1])) %42 = (22*t^9 + 16*t^7 + 45*t^6 + 82*t^5 + 59*t^4 + 11*t^3 + 50*t^2 + 11*t + 83)/(97*t^8 + 53*t^6 + 42*t^5 + 82*t^4 + 50*t^3 + 59*t^2 + 76*t + 65) ? ellpow(e,R,5) %43 = [Mod((Mod(24, 101)*t^25 + Mod(72, 101)*t^23 + Mod(72, 101)*t^22 + Mod(89, 101)*t^21 + Mod(16, 101)*t^20 + Mod(21, 101)*t^19 + Mod(85, 101)*t^18 + Mod(82, 101)*t^17 + Mod(11, 101)*t^16 + Mod(16, 101)*t^15 + Mod(40, 101)*t^14 + Mod(70, 101)*t^13 + Mod(84, 101)*t^12 + Mod(53, 101)*t^11 + Mod(72, 101)*t^10 + Mod(80, 101)*t^9 + Mod(85, 101)*t^8 + Mod(27, 101)*t^7 + Mod(60, 101)*t^6 + Mod(70, 101)*t^5 + Mod(75, 101)*t^4 + Mod(17, 101)*t^3 + Mod(16, 101)*t^2 + Mod(80, 101)*t + Mod(38, 101))/(Mod(95, 101)*t^24 + Mod(99, 101)*t^22 + Mod(86, 101)*t^21 + Mod(29, 101)*t^20 + Mod(94, 101)*t^19 + Mod(58, 101)*t^18 + Mod(5, 101)*t^17 + Mod(25, 101)*t^16 + Mod(89, 101)*t^15 + Mod(18, 101)*t^14 + Mod(91, 101)*t^13 + Mod(7, 101)*t^12 + Mod(96, 101)*t^11 + Mod(25, 101)*t^10 + Mod(49, 101)*t^9 + Mod(40, 101)*t^8 + Mod(66, 101)*t^7 + Mod(4, 101)*t^6 + Mod(13, 101)*t^5 + Mod(55, 101)*t^4 + Mod(79, 101)*t^3 + Mod(23, 101)*t^2 + Mod(39, 101)*t + Mod(5, 101)), Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101))), Mod(((Mod(54, 101)*t^36 + Mod(85, 101)*t^34 + Mod(98, 101)*t^33 + Mod(16, 101)*t^32 + Mod(9, 101)*t^31 + Mod(96, 101)*t^30 + Mod(12, 101)*t^29 + Mod(71, 101)*t^28 + Mod(95, 101)*t^27 + Mod(73, 101)*t^26 + Mod(89, 101)*t^25 + Mod(52, 101)*t^24 + Mod(4, 101)*t^23 + Mod(35, 101)*t^22 + Mod(86, 101)*t^21 + Mod(85, 101)*t^20 + Mod(14, 101)*t^19 + Mod(8, 101)*t^18 + Mod(71, 101)*t^17 + Mod(42, 101)*t^16 + Mod(46, 101)*t^15 + Mod(61, 101)*t^14 + Mod(17, 101)*t^13 + Mod(21, 101)*t^12 + Mod(93, 101)*t^11 + Mod(26, 101)*t^10 + Mod(90, 101)*t^9 + Mod(88, 101)*t^8 + Mod(93, 101)*t^7 + Mod(17, 101)*t^6 + Mod(25, 101)*t^5 + Mod(94, 101)*t^4 + Mod(85, 101)*t^3 + Mod(61, 101)*t^2 + Mod(67, 101)*t + Mod(32, 101))/(Mod(84, 101)*t^36 + Mod(42, 101)*t^34 + Mod(12, 101)*t^33 + Mod(51, 101)*t^32 + Mod(48, 101)*t^31 + Mod(99, 101)*t^30 + Mod(25, 101)*t^29 + Mod(94, 101)*t^28 + Mod(51, 101)*t^27 + Mod(30, 101)*t^26 + Mod(18, 101)*t^25 + Mod(98, 101)*t^24 + Mod(30, 101)*t^23 + Mod(22, 101)*t^22 + Mod(85, 101)*t^21 + Mod(36, 101)*t^20 + Mod(75, 101)*t^19 + Mod(57, 101)*t^18 + Mod(28, 101)*t^17 + Mod(63, 101)*t^16 + Mod(49, 101)*t^15 + Mod(73, 101)*t^14 + Mod(43, 101)*t^13 + Mod(67, 101)*t^12 + Mod(15, 101)*t^11 + Mod(49, 101)*t^10 + Mod(63, 101)*t^9 + Mod(2, 101)*t^8 + Mod(51, 101)*t^7 + Mod(15, 101)*t^6 + Mod(51, 101)*t^5 + Mod(41, 101)*t^4 + Mod(6, 101)*t^3 + Mod(83, 101)*t^2 + Mod(57, 101)*t + Mod(23, 101)))*x, Mod(1, 101)*x^2 + (Mod(100, 101)*t^3 + Mod(98, 101)*t + Mod(96, 101)))] ? lift(lift(%[1])) %44 = (24*t^25 + 72*t^23 + 72*t^22 + 89*t^21 + 16*t^20 + 21*t^19 + 85*t^18 + 82*t^17 + 11*t^16 + 16*t^15 + 40*t^14 + 70*t^13 + 84*t^12 + 53*t^11 + 72*t^10 + 80*t^9 + 85*t^8 + 27*t^7 + 60*t^6 + 70*t^5 + 75*t^4 + 17*t^3 + 16*t^2 + 80*t + 38)/(95*t^24 + 99*t^22 + 86*t^21 + 29*t^20 + 94*t^19 + 58*t^18 + 5*t^17 + 25*t^16 + 89*t^15 + 18*t^14 + 91*t^13 + 7*t^12 + 96*t^11 + 25*t^10 + 49*t^9 + 40*t^8 + 66*t^7 + 4*t^6 + 13*t^5 + 55*t^4 + 79*t^3 + 23*t^2 + 39*t + 5) ? \q Goodbye!