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TD (h)Tj /F2 1 Tf 0.5762 0 TD (\()Tj /F8 1 Tf 0.389 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD (\))Tj /F8 1 Tf 0.3889 0 TD 0.1667 Tc (,n)Tj /F2 1 Tf 1.0447 0 TD 0.4388 Tc [(\)t)438.8(o)]TJ -32.256 -1.2694 TD 0 Tc [(represen)27.9(t)-333.3(the)-333.4(equation)]TJ /F8 1 Tf 10.1222 0 TD (f)Tj /F2 1 Tf 0.5973 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD 0.2778 Tc (\)=)Tj /F8 1 Tf 1.7222 0 TD 0 Tc (g)Tj /F2 1 Tf 0.5129 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.907 0 TD [(\))-333.3(i)0(n)-333.3(the)-333.3(ring)]TJ /F8 1 Tf 5.6694 0 TD (Z)Tj /F11 1 Tf 7.9701 0 0 7.9701 401.599 590.5435 Tm (n)Tj /F2 1 Tf 10.9091 0 0 10.9091 407.2349 592.1793 Tm ([)Tj /F8 1 Tf 0.2778 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD (])Tj /F8 1 Tf 0.2778 0 TD (/)Tj /F2 1 Tf 0.5 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (h)Tj /F2 1 Tf 0.5762 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD (\)\).)Tj -24.6291 -1.2694 TD [(W)83.4(e)-503(use)-503(the)-503(sym)27.8(b)-27.8(o)0(l)]TJ /F8 1 Tf 9.2732 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 294.413 582.2904 Tm (
)Tj /F2 1 Tf 10.9091 0 0 10.9091 301.4974 578.3313 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (t)Tj /F2 1 Tf 0.3611 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD [(\)\))-502.9(for)]TJ /F8 1 Tf 2.981 0 TD (O)Tj /F2 1 Tf 0.7906 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (t)Tj /F2 1 Tf 0.3611 0 TD (\()Tj /F8 1 Tf 0.389 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD (\))Tj /F9 1 Tf 0.7242 0 TD (·)Tj /F2 1 Tf 0.6132 0 TD [(p)-27.8(oly\(log)]TJ /F8 1 Tf 3.7362 0 TD (t)Tj /F2 1 Tf 0.3611 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD [(\)\),)-545.5(where)]TJ /F8 1 Tf 4.6623 0 TD (t)Tj /F2 1 Tf 0.3612 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD 0.503 Tc [(\)i)503(s)]TJ -32.4085 -1.4429 TD 0 Tc [(an)27.8(y)-441.7(function)-441.6(of)]TJ /F8 1 Tf 7.2695 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD [(.)-769.5(F)83.4(or)-441.7(example,)]TJ /F8 1 Tf 7.28 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 336.6038 566.5494 Tm (
)Tj /F2 1 Tf 10.9091 0 0 10.9091 343.6882 562.5903 Tm (\(log)Tj /F11 1 Tf 7.9701 0 0 7.9701 362.0214 567.0117 Tm (k)Tj /F8 1 Tf 10.9091 0 0 10.9091 368.9594 562.5903 Tm (n)Tj /F2 1 Tf 0.6003 0 TD 0.4583 Tc (\)=)Tj /F8 1 Tf 2.0834 0 TD 0 Tc (O)Tj /F2 1 Tf 0.7906 0 TD (\(log)Tj /F11 1 Tf 7.9701 0 0 7.9701 425.1935 567.0117 Tm (k)Tj /F8 1 Tf 10.9091 0 0 10.9091 432.1315 562.5903 Tm (n)Tj /F9 1 Tf 0.8946 0 TD (·)Tj /F2 1 Tf 0.5722 0 TD [(p)-27.8(oly\(log)-180.6(log)]TJ /F8 1 Tf 5.1945 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD [(\)\))-458.4(=)]TJ /F8 1 Tf -31.9586 -1.2694 TD (O)Tj /F2 1 Tf 0.7905 0 TD (\(log)Tj /F11 1 Tf 7.9701 0 0 7.9701 189.6657 553.1636 Tm (k)Tj /F1 1 Tf 0.5799 0 TD (+)Tj /F11 1 Tf 0.8265 0 TD (
)Tj /F8 1 Tf 10.9091 0 0 10.9091 207.1014 548.7422 Tm (n)Tj /F2 1 Tf 0.6002 0 TD [(\))-290(for)-290.1(an)27.8(y)]TJ /F8 1 Tf 4.0118 0 TD 0.2778 Tc (
>)Tj /F2 1 Tf 1.7996 0 TD 0 Tc [(0.)-430(W)83.3(e)-290(use)-290(log)-303.9(for)-290(base)-290(2)-290(logarithms,)-298.5(and)-290(ln)-290(for)-290(natural)]TJ -10.4809 -1.2694 TD (logarithms.)Tj /F9 1 Tf 2.0091 -1.2694 TD (N)Tj /F2 1 Tf 1.4317 0 TD (and)Tj /F9 1 Tf 2.075 0 TD (Z)Tj /F2 1 Tf 1.268 0 TD [(denote)-463.8(the)-463.9(set)-463.8(of)-463.8(natural)-463.8(n)27.8(u)0(m)27.8(b)-27.8(ers)-463.9(and)-463.8(in)27.8(tegers)-463.8(resp)-27.7(ectiv)27.9(ely)83.4(.)]TJ -6.7838 -1.2694 TD [(Giv)27.8(en)]TJ /F8 1 Tf 2.9406 0 TD (r)Tj /F9 1 Tf 0.8314 0 TD 0.3524 Tc (N)Tj /F2 1 Tf 1.9869 0 TD 0 Tc (,)Tj /F8 1 Tf 0.6672 0 TD (a)Tj /F9 1 Tf 0.881 0 TD 0.3525 Tc (Z)Tj /F2 1 Tf 2.2015 0 TD 0 Tc [(with)-378.2(\()]TJ /F8 1 Tf 2.7115 0 TD [(a,)-166.7(r)]TJ /F2 1 Tf 1.452 0 TD [(\))-352.5(=)-352.4(1,)-389.3(the)]TJ /F10 1 Tf 4.8058 0 TD [(or)51.1(der)-399(of)]TJ /F8 1 Tf 3.8902 0 TD (a)Tj /F10 1 Tf 0.9276 0 TD [(mo)51.1(dulo)]TJ /F8 1 Tf 3.4912 0 TD (r)Tj /F2 1 Tf 0.8571 0 TD [(is)-378.2(the)-378.1(smallest)]TJ -27.644 -1.2694 TD -0.0278 Tc [(nu)-27.8(mb)-55.6(e)-27.8(r)]TJ /F8 1 Tf 3.6709 0 TD 0 Tc (k)Tj /F2 1 Tf 0.9146 0 TD [(suc)27.8(h)-362.5(that)]TJ /F8 1 Tf 4.4805 0 TD (a)Tj /F11 1 Tf 7.9701 0 0 7.9701 267.3783 497.3091 Tm (k)Tj /F2 1 Tf 10.9091 0 0 10.9091 276.0588 493.3499 Tm 0.3264 Tc [(=1\()326.3(m)326.4(o)298.7(d)]TJ /F8 1 Tf 4.5695 0 TD 0 Tc (r)Tj /F2 1 Tf 0.479 0 TD [(\).)-532(It)-362.5(is)-362.6(denoted)-362.4(as)-362.5(o)]TJ /F11 1 Tf 7.9701 0 0 7.9701 428.3318 491.7141 Tm (r)Tj /F2 1 Tf 10.9091 0 0 10.9091 432.8867 493.3499 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (a)Tj /F2 1 Tf 0.5286 0 TD 0.0001 Tc [(\).)-531.9(F)83.4(o)0.1(r)]TJ /F8 1 Tf 3.0224 0 TD 0 Tc (r)Tj /F9 1 Tf 0.8054 0 TD 0.3264 Tc (N)Tj /F2 1 Tf 1.9609 0 TD 0 Tc (,)Tj /F8 1 Tf 0.6476 0 TD ()Tj /F2 1 Tf 0.5959 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (r)Tj /F2 1 Tf 0.4789 0 TD (\))Tj -33.5837 -1.2694 TD (is)Tj /F10 1 Tf 1.0271 0 TD (Euler)Tj /F2 1 Tf 2.4598 0 TD ()Tj /F10 1 Tf 0.2777 0 TD [(s)-377.6(totient)-377.5(function)]TJ /F2 1 Tf 8.0101 0 TD [(giving)-354.7(the)-354.9(n)27.8(u)0(m)27.8(b)-27.7(er)-354.9(of)-354.9(n)27.8(u)0(m)27.8(b)-27.7(ers)-354.9(less)-354.8(than)]TJ /F8 1 Tf 17.8396 0 TD (r)Tj /F2 1 Tf 0.8339 0 TD [(that)-354.7(are)]TJ -30.4483 -1.2694 TD [(relativ)27.9(ely)-333.3(prime)-333.3(to)]TJ /F8 1 Tf 8.4222 0 TD (r)Tj /F2 1 Tf 0.479 0 TD [(.)-444.4(I)0(t)-333.3(i)0(s)-333.3(easy)-333.3(to)-333.4(see)-333.3(that)-333.3(o)]TJ /F11 1 Tf 7.9701 0 0 7.9701 374.5395 464.018 Tm (r)Tj /F2 1 Tf 10.9091 0 0 10.9091 379.0944 465.6538 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (a)Tj /F2 1 Tf 0.5286 0 TD (\))Tj /F9 1 Tf 0.6667 0 TD (|)Tj /F8 1 Tf 0.5556 0 TD ()Tj /F2 1 Tf 0.5958 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (r)Tj /F2 1 Tf 0.479 0 TD [(\))-333.3(for)-333.3(an)27.8(y)]TJ /F8 1 Tf 4.1416 0 TD (a)Tj /F2 1 Tf 0.5286 0 TD 0.3334 Tc (,\()Tj /F8 1 Tf 1 0 TD 0 Tc [(a,)-166.7(r)]TJ /F2 1 Tf 1.452 0 TD 0.2778 Tc [(\)=1)277.8(.)]TJ -28.5518 -1.2694 TD 0 Tc [(W)83.4(e)-238.7(will)-238.7(need)-238.6(the)-238.6(follo)27.8(wing)-238.6(simple)-238.6(fact)-238.6(ab)-27.7(out)-238.6(the)-238.6(lcm)-238.7(of)-238.6(the)-238.6(“rst)]TJ /F8 1 Tf 27.144 0 TD (m)Tj /F2 1 Tf 1.1166 0 TD [(n)27.7(u)0(m)27.8(b)-27.7(ers)]TJ -30.2698 -1.2694 TD [(\(see,)-333.2(e.g.,)-333.3([Nai])-333.4(for)-333.3(a)-333.3(pro)-27.7(of)-77.8(\).)]TJ /F6 1 Tf 2.0091 -3.0418 TD [(Lemma)-377.6(3.1.)]TJ /F10 1 Tf 5.97 0 TD -0.0511 Tc [(Le)-51.1(t)]TJ /F2 1 Tf 1.8239 0 TD 0 Tc (LCM\()Tj /F8 1 Tf 2.6528 0 TD (m)Tj /F2 1 Tf 0.878 0 TD (\))Tj /F10 1 Tf 0.8446 0 TD [(denote)-455.5(the)]TJ /F2 1 Tf 5.0512 0 TD (lcm)Tj /F10 1 Tf 2.0111 0 TD [(of)-455.6(the)-455.6(“rst)]TJ /F8 1 Tf 5.213 0 TD (m)Tj /F10 1 Tf 1.3337 0 TD [(numb)51.1(ers.)-753.5(F)76.7(o)0(r)]TJ /F8 1 Tf -27.7874 -1.2694 TD (m)Tj /F9 1 Tf 1.1558 0 TD ()Tj /F2 1 Tf 1.0555 0 TD (7:)Tj 11.3934 -3.1648 TD (LCM\()Tj /F8 1 Tf 2.6528 0 TD (m)Tj /F2 1 Tf 0.878 0 TD (\))Tj /F9 1 Tf 0.6667 0 TD ()Tj /F2 1 Tf 1.0555 0 TD (2)Tj /F11 1 Tf 7.9701 0 0 7.9701 373.8839 360.9066 Tm (m)Tj /F8 1 Tf 10.9091 0 0 10.9091 381.873 356.4015 Tm (.)Tj /F7 1 Tf -12.3456 -3.7633 TD [(4.)-574.9(The)-383.3(algorithm)-383.1(and)-383.3(its)-383.3(correctness)]TJ ET 162.69 296.529 387.632 -0.398 re f 162.45 296.133 0.399 -18.003 re f BT /F18 1 Tf 10.9091 0 0 10.9091 168.6864 283.5178 Tm [(Input:)-1049.8(integer)]TJ /F8 1 Tf 8.3999 0 TD 0.2778 Tc (n>)Tj /F2 1 Tf 1.9335 0 TD 0 Tc (1)Tj /F18 1 Tf 0.5 0 TD (.)Tj ET 550.17 296.133 0.398 -18.003 re f 162.45 278.133 0.399 -18.002 re f BT 10.9091 0 0 10.9091 168.6864 265.5154 Tm [(1.)-1095.8(If)-525(\()]TJ /F8 1 Tf 4.2458 0 TD (n)Tj /F2 1 Tf 0.878 0 TD (=)Tj /F8 1 Tf 1.0555 0 TD (a)Tj /F11 1 Tf 7.9701 0 0 7.9701 241.865 269.4745 Tm (b)Tj /F18 1 Tf 10.9091 0 0 10.9091 251.7121 265.5154 Tm (for)Tj /F8 1 Tf 2.1 0 TD (a)Tj /F9 1 Tf 0.8064 0 TD 0.2777 Tc (N)Tj /F18 1 Tf 2.4373 0 TD 0 Tc (and)Tj /F8 1 Tf 2.1 0 TD 0.2778 Tc (b>)Tj /F2 1 Tf 1.7625 0 TD 0 Tc (1)Tj /F18 1 Tf 0.5 0 TD [(\),)-525(output)-524.9(COMPOSITE.)]TJ ET 550.17 278.133 0.398 -18.002 re f 162.45 260.133 0.399 -18.002 re f BT 10.9091 0 0 10.9091 168.6864 247.5128 Tm [(2.)-1095.8(Find)-525(the)-524.9(smallest)]TJ /F8 1 Tf 11.5958 0 TD (r)Tj /F18 1 Tf 1.0039 0 TD [(such)-524.9(that)-525(o)]TJ /F11 1 Tf 7.9701 0 0 7.9701 369.1367 245.876 Tm (r)Tj /F2 1 Tf 10.9091 0 0 10.9091 373.6917 247.5128 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD (\))Tj /F8 1 Tf 0.6667 0 TD (>)Tj /F2 1 Tf 1.0555 0 TD (log)Tj /F1 1 Tf 7.9701 0 0 7.9701 417.3609 251.9332 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 423.9113 247.5128 Tm (n)Tj /F18 1 Tf 0.6003 0 TD (.)Tj ET 550.17 260.133 0.398 -18.002 re f 162.45 242.133 0.399 -18.002 re f BT 10.9091 0 0 10.9091 168.6864 229.5103 Tm [(3.)-1095.8(If)]TJ /F2 1 Tf 3.7208 0 TD (1)Tj /F8 1 Tf 0.7778 0 TD (<)Tj /F2 1 Tf 1.0555 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD [(a,)-166.6(n)]TJ /F2 1 Tf 1.5732 0 TD (\))Tj /F8 1 Tf 0.6667 0 TD 0.2777 Tc (> /ExtGState << /GS1 17 0 R >> >> endobj 34 0 obj << /Length 15379 >> stream BT /F6 1 Tf 7.9701 0 0 7.9701 217.054 726.9739 Tm 0 g /GS1 gs 0 Tc 0 Tw [(PRIMES)-377.6(IS)-377.8(IN)]TJ /F1 1 Tf 7.6583 0 TD (P)Tj /F2 1 Tf 10.9091 0 0 10.9091 429.081 726.974 Tm (785)Tj /F6 1 Tf -30.4635 -2.3744 TD [(Theorem)-377.7(4.1.)]TJ /F10 1 Tf 7.1216 0 TD [(The)-426.8(algorithm)-426.8(ab)51.2(ove)-426.9(r)51.1(eturns)]TJ /F2 1 Tf 12.9646 0 TD (PRIME)Tj /F10 1 Tf 3.8017 0 TD [(if)-426.8(and)-426.8(only)-426.8(if)]TJ /F8 1 Tf 6.3329 0 TD (n)Tj /F10 1 Tf 1.0271 0 TD (is)Tj -33.2571 -1.2694 TD (prime.)Tj /F2 1 Tf 2.0091 -2.1395 TD [(In)-379.3(the)-379.3(remainder)-379.2(of)-379.4(the)-379.3(section,)-390.7(w)27.8(e)-379.3(establish)-379.2(this)-379.4(theorem)-379.2(through)-379.3(a)-379.3(se-)]TJ -2.0091 -1.2694 TD [(quence)-333.3(of)-333.3(lemmas.)-444.4(The)-333.3(follo)27.8(wing)-333.3(is)-333.2(trivial:)]TJ /F6 1 Tf 2.0091 -2.1395 TD [(Lemma)-377.7(4.2.)]TJ /F10 1 Tf 5.97 0 TD (If)Tj /F8 1 Tf 1.05 0 TD (n)Tj /F10 1 Tf 0.958 0 TD [(is)-357.8(prime)]TJ /F2 1 Tf 3.5905 0 TD (,)Tj /F10 1 Tf 0.6356 0 TD [(the)-357.7(algorithm)-357.6(r)51.2(eturns)]TJ /F2 1 Tf 9.596 0 TD (PRIME)Tj /F10 1 Tf 3.375 0 TD (.)Tj -25.1751 -2.1395 TD [(Pr)51.1(o)51.2(o)0(f)]TJ /F2 1 Tf 2.3267 0 TD 0.5851 Tc [(.I)585.1(f)]TJ /F8 1 Tf 1.9098 0 TD 0 Tc (n)Tj /F2 1 Tf 0.9805 0 TD [(is)-380.2(prime)-380.2(then)-380.2(steps)-380.2(1)-380.2(and)-380.2(3)-380.3(can)-380.2(nev)27.8(er)-380.2(return)-380.1(COMPOSITE.)]TJ -7.226 -1.2694 TD [(By)-337.6(Lemma)-337.5(2.1,)-338.7(the)]TJ /F18 1 Tf 8.768 0 TD (for)Tj /F2 1 Tf 1.9125 0 TD [(lo)-27.8(op)-337.6(also)-337.5(cannot)-337.5(return)-337.5(COMPOSITE.)-337.5(Therefore)-337.4(the)]TJ -10.6805 -1.2694 TD [(algorithm)-333.3(will)-333.3(iden)27.8(tify)]TJ /F8 1 Tf 10.0861 0 TD (n)Tj /F2 1 Tf 0.9336 0 TD [(as)-333.3(PRIME)-333.2(either)-333.3(in)-333.3(step)-333.3(4)-333.3(o)0(r)-333.3(i)0(n)-333.3(step)-333.3(6.)]TJ ET 438.69 582.489 6.775 -0.398 re f 438.69 582.068 0.399 -5.978 re f 445.05 582.068 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)Tj /F2 1 Tf 10.9091 0 0 10.9091 387.8247 381.5293 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (m)Tj /F2 1 Tf 0.8781 0 TD [(\))-251([vzGG].)]TJ -29.9577 -1.2694 TD [(Similarly)83.4(,)-462.6(these)-436.7(op)-27.8(erations)-436.7(on)-436.8(t)27.8(w)27.8(o)-436.8(degree)]TJ /F8 1 Tf 18.9161 0 TD (d)Tj /F2 1 Tf 0.9573 0 TD [(p)-27.8(olynomials)-436.6(with)-436.8(co)-27.8(ecien)27.7(ts)-436.7(at)]TJ -19.8733 -1.2694 TD (most)Tj /F8 1 Tf 2.45 0 TD (m)Tj /F2 1 Tf 1.2113 0 TD [(bits)-333.3(in)-333.3(size)-333.3(can)-333.3(b)-27.7(e)-333.4(done)-333.3(in)-333.3(time)]TJ /F8 1 Tf 14.0389 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 276.5529 357.7913 Tm (
)Tj /F2 1 Tf 10.9091 0 0 10.9091 283.6374 353.8331 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (d)Tj /F9 1 Tf 0.7427 0 TD (·)Tj /F8 1 Tf 0.5 0 TD (m)Tj /F2 1 Tf 0.878 0 TD [(\))-333.3(steps)-333.3([vzGG].)]TJ /F6 1 Tf -19.6407 -2.1352 TD [(Theorem)-377.7(5.1.)]TJ /F10 1 Tf 7.1216 0 TD [(The)-720.7(asymptotic)-720.7(time)-720.7(c)51.1(omplexity)-720.6(of)-720.7(the)-720.8(algorithm)-720.6(is)]TJ /F8 1 Tf -9.1308 -1.2694 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 83.459 320.6516 Tm (
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)Tj /F2 1 Tf 10.9091 0 0 10.9091 133.6946 196.4633 Tm (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 152.0279 200.8837 Tm (7)Tj /F8 1 Tf 10.9091 0 0 10.9091 158.5793 196.4633 Tm (n)Tj /F2 1 Tf 0.6002 0 TD (\).)Tj -6.2677 -1.2694 TD [(The)-255.3(third)-255.2(step)-255.3(in)27.9(v)27.8(olv)27.8(es)-255.3(computing)-255.2(the)-255.2(gcd)-255.2(of)]TJ /F8 1 Tf 19.4448 0 TD (r)Tj /F2 1 Tf 0.7342 0 TD [(n)27.8(u)0(m)27.7(b)-27.7(ers.)-418.4(Eac)27.8(h)-255.2(gcd)-255.2(compu-)]TJ -22.1881 -1.2694 TD [(tation)-355.9(tak)27.8(es)-355.9(time)]TJ /F8 1 Tf 7.851 0 TD (O)Tj /F2 1 Tf 0.7906 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD [(\))-355.9([vzGG],)-361.5(and)-355.9(therefore,)-361.5(the)-355.8(time)-355.9(complexit)27.9(y)-356(o)0(f)-355.9(this)]TJ -11.089 -1.2694 TD [(step)-333.3(is)]TJ /F8 1 Tf 3.1223 0 TD (O)Tj /F2 1 Tf 0.7905 0 TD (\()Tj /F8 1 Tf 0.389 0 TD (r)Tj /F2 1 Tf 0.6456 0 TD (log)Tj /F8 1 Tf 1.4584 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD 0.2778 Tc (\)=)Tj /F8 1 Tf 1.7222 0 TD 0 Tc (O)Tj /F2 1 Tf 0.7906 0 TD (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 197.0072 159.3394 Tm (6)Tj /F8 1 Tf 10.9091 0 0 10.9091 203.5576 154.919 Tm (n)Tj /F2 1 Tf 0.6003 0 TD [(\).)-444.4(The)-333.4(time)-333.2(complexit)27.8(y)-333.3(o)0(f)-333.3(step)-333.2(4)-333.3(i)0(s)-333.3(just)]TJ /F8 1 Tf 17.6 0 TD (O)Tj /F2 1 Tf 0.7905 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD (\).)Tj -31.2288 -1.2694 TD [(In)-391.6(step)-391.5(5,)-406.1(w)27.8(e)-391.5(need)-391.6(to)-391.6(v)27.8(erify)]TJ /F9 1 Tf 12.7084 0 TD ()Tj /F14 1 Tf 0.4445 0.855 TD ()Tj ET 251.13 150.886 20.209 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 251.1472 141.071 Tm ()Tj /F2 1 Tf 0.5959 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (r)Tj /F2 1 Tf 0.4789 0 TD [(\))-166.7(log)]TJ /F8 1 Tf 2.0139 0 TD (n)Tj /F9 1 Tf 0.6003 0 TD ( )Tj /F2 1 Tf 0.836 0 TD [(equations.)-619.1(Eac)27.9(h)-391.7(equation)-391.5(re-)]TJ -21.0758 -1.2694 TD (quires)Tj /F8 1 Tf 3.0923 0 TD (O)Tj /F2 1 Tf 0.7905 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD [(\))-500.6(m)27.8(ultiplications)-500.5(of)-500.6(degree)]TJ /F8 1 Tf 12.4553 0 TD (r)Tj /F2 1 Tf 0.9795 0 TD [(p)-27.7(olynomials)-500.5(with)-500.5(co)-27.8(ecien)27.8(ts)-500.6(of)]TJ ET endstream endobj 50 0 obj << /ProcSet [/PDF /Text ] /Font << /F1 4 0 R /F2 5 0 R /F6 9 0 R /F7 10 0 R /F8 11 0 R /F9 12 0 R /F10 13 0 R /F11 14 0 R /F12 15 0 R /F14 22 0 R >> /ExtGState << /GS1 17 0 R >> >> endobj 52 0 obj << /Length 16592 >> stream BT /F2 1 Tf 10.9091 0 0 10.9091 162.7088 726.9739 Tm 0 g /GS1 gs 0 Tc 0 Tw (790)Tj /F6 1 Tf 7.9701 0 0 7.9701 223.2349 726.9739 Tm [(MANINDRA)-377.5(A)29.3(GRA)116.7(W)116.7(AL,)-377.6(NEERAJ)-377.5(KA)87.5(Y)87.5(AL,)-377.6(AND)-377.7(NITIN)-377.7(SAXENA)]TJ /F2 1 Tf 10.9091 0 0 10.9091 162.7088 701.071 Tm (size)Tj /F8 1 Tf 2.0268 0 TD (O)Tj /F2 1 Tf 0.7906 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD [(\).)-841.5(So)-465.6(eac)27.8(h)-465.7(equation)-465.6(can)-465.6(b)-27.7(e)-465.7(v)27.8(eri“ed)-465.6(in)-465.7(time)]TJ /F8 1 Tf 20.4309 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 451.6494 705.0302 Tm (
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)Tj /F2 1 Tf 10.9091 0 0 10.9091 365.6419 687.2229 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (r)Tj /F14 1 Tf 0.4789 0.855 TD ()Tj ET 386.01 697.006 20.209 -0.436 re f BT /F8 1 Tf 10.9091 0 0 10.9091 386.0175 687.2229 Tm ()Tj /F2 1 Tf 0.5959 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (r)Tj /F2 1 Tf 0.479 0 TD [(\))-166.6(log)]TJ /F1 1 Tf 7.9701 0 0 7.9701 422.137 691.6434 Tm (3)Tj /F8 1 Tf 10.9091 0 0 10.9091 428.6874 687.2229 Tm (n)Tj /F2 1 Tf 0.6002 0 TD 0.3717 Tc (\)=)Tj /F8 1 Tf 1.9102 0 TD 0 Tc (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 464.6974 691.1821 Tm (
)Tj /F2 1 Tf 10.9091 0 0 10.9091 471.7819 687.2229 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (r)Tj /F17 1 Tf 4.9813 0 0 4.9813 482.4449 694.7816 Tm (3)Tj ET 482.49 693.703 3.39 -0.373 re f BT 4.9813 0 0 4.9813 482.4449 688.3975 Tm (2)Tj /F2 1 Tf 10.9091 0 0 10.9091 489.3469 687.2229 Tm (log)Tj /F1 1 Tf 7.9701 0 0 7.9701 503.4381 691.6434 Tm (3)Tj /F8 1 Tf 10.9091 0 0 10.9091 509.9886 687.2229 Tm (n)Tj /F2 1 Tf 0.6002 0 TD 0.3717 Tc (\)=)Tj /F8 1 Tf -32.4342 -1.2694 TD 0 Tc (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 171.3325 677.334 Tm (
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)Tj /F2 1 Tf 10.9091 0 0 10.9091 228.6814 593.5303 Tm (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 247.0147 597.9517 Tm (6)Tj /F8 1 Tf 10.9091 0 0 10.9091 253.5651 593.5303 Tm (n)Tj /F2 1 Tf 0.6003 0 TD [(\).)-939.1(In)-498.2(fact,)-539.4(there)-498.1(are)-498.2(t)27.8(w)27.7(o)-498.2(conjectures)-498.1(that)-498.1(supp)-27.8(ort)-498.1(the)]TJ -8.9288 -1.2694 TD [(p)-27.7(ossibilit)27.8(y)-333.3(o)0(f)-333.3(suc)27.8(h)-333.3(a)0(n)]TJ /F8 1 Tf 9.5445 0 TD (r)Tj /F2 1 Tf 0.8122 0 TD [(\(b)-27.8(elo)27.8(w)-333.3(l)0(n)-333.3(i)0(s)-333.3(the)-333.3(natural)-333.3(logarithm\):)]TJ /F10 1 Tf -8.3476 -2.1827 TD [(A)25.6(rtin)]TJ /F2 1 Tf 2.4173 0 TD ()Tj /F10 1 Tf 0.2777 0 TD [(s)-475.5(Conje)51.1(ctur)51.2(e)]TJ /F2 1 Tf 5.5482 0 TD [(.)-1656.5(Giv)27.8(en)-461.2(an)27.7(y)-461.3(n)27.8(um)27.8(b)-27.7(e)0(r)]TJ /F8 1 Tf 10.7445 0 TD (n)Tj /F9 1 Tf 1.0911 0 TD 0.491 Tc (N)Tj /F2 1 Tf 2.5868 0 TD 0 Tc [(that)-461.1(is)-461.3(not)-461.3(a)-461.2(p)-27.8(erfect)]TJ -24.6748 -1.2694 TD [(square,)-323.2(the)-320.9(n)27.8(u)0(m)27.7(b)-27.7(er)-320.9(of)-320.8(primes)]TJ /F8 1 Tf 13.0985 0 TD (q)Tj /F9 1 Tf 0.76 0 TD (
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