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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 0.398 -5.978 re f 438.69 576.009 6.775 -0.398 re f BT 10.9091 0 0 10.9091 96.7522 553.2314 Tm [(The)-497.6(con)27.8(v)27.8(erse)-497.6(of)-497.5(the)-497.6(ab)-27.8(o)27.8(v)27.8(e)-497.6(lemma)-497.5(requires)-497.5(a)-497.5(little)-497.5(more)-497.5(w)27.8(ork.)-937.2(If)-497.6(the)]TJ -2.0091 -1.2694 TD [(algorithm)-473.1(returns)-473.1(PRIME)-473(in)-473.1(step)-473.2(4)-473.2(then)]TJ /F8 1 Tf 19.0954 0 TD (n)Tj /F2 1 Tf 1.0734 0 TD [(m)27.9(ust)-473.2(b)-27.7(e)-473.2(prime)-473.1(since)-473.1(otherwise)]TJ -20.1689 -1.2694 TD [(step)-299.5(3)-299.4(w)27.8(ould)-299.4(ha)27.7(v)27.8(e)-299.4(found)-299.4(a)-299.4(non)27.8(trivial)-299.4(factor)-299.3(of)]TJ /F8 1 Tf 20.0671 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD [(.)-433.2(S)0(o)-299.4(the)-299.4(only)-299.4(remaining)-299.3(case)-299.4(is)]TJ -20.6673 -1.2694 TD [(when)-348.2(the)-348.2(algorithm)-348.1(returns)-348.1(PRIME)-348.2(in)-348.2(step)-348.2(6.)-489.1(F)83.4(o)0(r)-348.2(the)-348.2(purp)-27.7(ose)-348.2(of)-348.2(subsequen)27.9(t)]TJ T* [(analysis)-333.2(w)27.7(e)-333.3(assume)-333.3(this)-333.3(to)-333.4(b)-27.7(e)-333.4(the)-333.3(case.)]TJ 2.0091 -1.2694 TD [(The)-340(algorithm)-339.9(has)-340(t)27.7(w)27.8(o)-340(main)-340(steps)-340(\(2)-340(and)-340(5\):)-457.9(step)-340(2)-340(nds)-340(an)-340(appropriate)]TJ 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[(b)-27.8(eha)27.8(v)27.8(e)-343.1(lik)27.8(e)-343.2(prime)]TJ /F8 1 Tf 8.0323 0 TD (p)Tj /F2 1 Tf 0.8462 0 TD [(in)-343.2(the)-343.1(ab)-27.7(o)27.8(v)27.8(e)-343.2(equation.)]TJ -24.1931 -1.2694 TD [(W)83.4(e)-333.4(giv)27.8(e)-333.3(a)-333.3(name)-333.4(to)-333.3(this)-333.3(prop)-27.8(ert)27.8(y:)]TJ /F10 1 Tf 2.0091 -2.1764 TD (Denition)Tj /F6 1 Tf 4.7361 0 TD (4.4.)Tj /F2 1 Tf 2.2444 0 TD [(F)83.3(o)0(r)-321.6(p)-27.7(olynomial)]TJ /F8 1 Tf 6.9376 0 TD (f)Tj /F2 1 Tf 0.5973 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD [(\))-321.6(and)-321.5(n)27.8(u)0(m)27.8(b)-27.8(er)]TJ /F8 1 Tf 6.2731 0 TD (m)Tj /F9 1 Tf 1.1558 0 TD 0.2778 Tc (N)Tj /F2 1 Tf 1.9123 0 TD 0 Tc [(,)-324(w)27.8(e)-321.6(s)0(a)27.8(y)-321.5(that)]TJ /F8 1 Tf 5.9331 0 TD (m)Tj /F2 1 Tf -33.0946 -1.2694 TD (is)Tj /F10 1 Tf 1.0056 0 TD [(intr)51.2(osp)51.1(e)51.1(ctive)]TJ /F2 1 Tf 5.7879 0 TD (for)Tj /F8 1 Tf 1.5305 0 TD (f)Tj /F2 1 Tf 0.5973 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD 0.3334 Tc [(\)i)333.4(f)]TJ -1.0059 -1.8769 TD 0 Tc ([)Tj /F8 1 Tf 0.2777 0 TD (f)Tj /F2 1 Tf 0.5973 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD (\)])Tj /F11 1 Tf 7.9701 0 0 7.9701 294.1469 442.1669 Tm (m)Tj /F2 1 Tf 10.9091 0 0 10.9091 305.1666 437.6628 Tm (=)Tj /F8 1 Tf 1.0555 0 TD (f)Tj /F2 1 Tf 0.5973 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F11 1 Tf 7.9701 0 0 7.9701 337.333 442.1669 Tm (m)Tj /F2 1 Tf 10.9091 0 0 10.9091 345.322 437.6628 Tm [(\))-333.3(\(mo)-27.7(d)]TJ /F8 1 Tf 3.3611 0 TD (X)Tj /F11 1 Tf 7.9701 0 0 7.9701 391.8824 441.6219 Tm (r)Tj /F9 1 Tf 10.9091 0 0 10.9091 398.8613 437.6628 Tm ()Tj /F2 1 Tf 1 0 TD (1)Tj /F8 1 Tf 0.5 0 TD 0.1667 Tc (,p)Tj /F2 1 Tf 0.9476 0 TD 0 Tc (\))Tj /F8 1 Tf 0.3889 0 TD (.)Tj /F2 1 Tf -22.4746 -2.1764 TD [(It)-326.8(is)-326.9(clear)-326.7(from)-326.8(equations)-326.7(\(5\))-326.8(and)-326.8(\(4\))-326.8(that)-326.8(b)-27.7(oth)]TJ /F11 1 Tf 7.9701 0 0 7.9701 414.3861 418.2147 Tm (n)Tj ET 414.33 416.806 5.138 -0.436 re f BT 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[(ar)51.1(e)-395.1(intr)51.2(osp)51.1(e)51.1(ctive)-394.9(numb)51.1(ers)-395(for)]TJ /F8 1 Tf 13.1566 0 TD (f)Tj /F2 1 Tf 0.5972 0 TD (\()Tj /F8 1 Tf 0.389 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD (\))Tj /F10 1 Tf 0.7838 0 TD [(then)-395(so)-395.1(is)]TJ /F8 1 Tf -29.6815 -1.2694 TD (m)Tj /F9 1 Tf 1.1002 0 TD ()Tj /F8 1 Tf 0.5 0 TD (m)Tj /F12 1 Tf 7.9701 0 0 7.9701 189.7444 338.7427 Tm ()Tj /F10 1 Tf 10.9091 0 0 10.9091 192.539 334.7835 Tm (.)Tj -0.7253 -2.1764 TD [(Pr)51.2(o)51.1(o)0(f)]TJ /F2 1 Tf 2.3267 0 TD [(.)-444.4(Since)]TJ /F8 1 Tf 3.3333 0 TD (m)Tj /F2 1 Tf 1.2113 0 TD [(is)-333.3(in)27.8(trosp)-27.7(ectiv)27.8(e)-333.3(for)]TJ /F8 1 Tf 8.4333 0 TD (f)Tj /F2 1 Tf 0.5973 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD -0.0278 Tc [(\))-361.1(w)0(e)-361.2(h)-27.8(ave)-27.8(:)]TJ -11.3379 -1.8768 TD 0 Tc ([)Tj /F8 1 Tf 0.2778 0 TD (f)Tj /F2 1 Tf 0.5972 0 TD (\()Tj /F8 1 Tf 0.389 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD (\)])Tj /F11 1 Tf 7.9701 0 0 7.9701 279.5069 295.0704 Tm (m)Tj /F12 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[(is)-329.6(in)27.8(trosp)-27.8(ectiv)27.9(e)-329.6(for)]TJ /F8 1 Tf 8.4222 0 TD (f)Tj /F2 1 Tf 0.5972 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD [(\),)-330.3(w)27.8(e)-329.6(ha)27.8(v)27.8(e)-329.6(\(after)-329.5(replacing)]TJ /F8 1 Tf 11.7932 0 TD (X)Tj /F2 1 Tf 1.2365 0 TD -0.0278 Tc (by)Tj /F8 1 Tf 1.3851 0 TD 0 Tc (X)Tj /F11 1 Tf 7.9701 0 0 7.9701 512.6436 274.0503 Tm (m)Tj /F2 1 Tf 10.9091 0 0 10.9091 524.2281 270.0911 Tm (in)Tj -33.1393 -1.2694 TD [(the)-333.3(in)27.8(trosp)-27.8(ection)-333.2(equation)-333.3(for)]TJ /F8 1 Tf 13.3445 0 TD (m)Tj /F12 1 Tf 7.9701 0 0 7.9701 317.863 260.2022 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 320.6576 256.243 Tm (\):)Tj -13.3214 -1.8768 TD ([)Tj /F8 1 Tf 0.2777 0 TD (f)Tj /F2 1 Tf 0.5973 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (X)Tj /F11 1 Tf 7.9701 0 0 7.9701 199.0147 240.2729 Tm (m)Tj /F2 1 Tf 10.9091 0 0 10.9091 207.0037 235.7688 Tm (\)])Tj /F11 1 Tf 7.9701 0 0 7.9701 214.2765 240.2729 Tm (m)Tj /F16 1 Tf 4.9813 0 0 4.9813 221.7664 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[(\))-333.3(\(mo)-27.8(d)]TJ /F8 1 Tf 3.3611 0 TD (X)Tj /F11 1 Tf 7.9701 0 0 7.9701 324.8199 222.8902 Tm (r)Tj /F9 1 Tf 10.9091 0 0 10.9091 331.7988 218.932 Tm ()Tj /F2 1 Tf 1 0 TD (1)Tj /F8 1 Tf 0.5 0 TD 0.1667 Tc (,p)Tj /F2 1 Tf 0.9476 0 TD 0 Tc [(\))-277.8(\(since)]TJ /F8 1 Tf 3.5056 0 TD (X)Tj /F11 1 Tf 7.9701 0 0 7.9701 406.6361 222.8902 Tm (r)Tj /F9 1 Tf 10.9091 0 0 10.9091 413.615 218.932 Tm ()Tj /F2 1 Tf 1 0 TD [(1)-333.3(divides)]TJ /F8 1 Tf 4.2 0 TD (X)Tj /F11 1 Tf 7.9701 0 0 7.9701 480.2361 222.8902 Tm (m)Tj /F12 1 Tf 0.9399 0 TD ()Tj /F11 1 Tf 0.2951 0 TD (r)Tj /F9 1 Tf 10.9091 0 0 10.9091 497.058 218.932 Tm ()Tj /F2 1 Tf 1 0 TD (1\))Tj /F8 1 Tf 0.8889 0 TD (.)Tj /F2 1 Tf -32.5375 -1.8769 TD [(Putting)-333.2(together)-333.2(the)-333.3(ab)-27.8(o)27.8(v)27.8(e)-333.3(t)27.7(w)27.8(o)-333.3(equations)-333.2(w)27.7(e)-333.3(get:)]TJ 8.5891 -2.3653 TD ([)Tj /F8 1 Tf 0.2778 0 TD (f)Tj /F2 1 Tf 0.5972 0 TD (\()Tj /F8 1 Tf 0.389 0 TD (X)Tj /F2 1 Tf 0.9069 0 TD (\)])Tj /F11 1 Tf 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(k>)Tj /F2 1 Tf 1.9731 0 TD 0 Tc [(1)-359.6(then)-359.6(the)-359.6(algorithm)-359.6(will)-359.6(return)-359.6(COMPOSITE)-359.5(in)-359.6(step)-359.6(1.)]TJ -8.8054 -1.2694 TD (Therefore,)Tj /F8 1 Tf 4.8111 0 TD (n)Tj /F2 1 Tf 0.878 0 TD (=)Tj /F8 1 Tf 1.0555 0 TD (p)Tj /F2 1 Tf 0.5032 0 TD (.)Tj ET 438.69 495.849 6.775 -0.398 re f 438.69 495.428 0.399 -5.978 re f 445.05 495.428 0.398 -5.978 re f 438.69 489.489 6.775 -0.398 re f BT 10.9091 0 0 10.9091 96.7522 466.8094 Tm [(This)-333.3(completes)-333.2(the)-333.4(pro)-27.7(of)-333.3(of)-333.3(Theorem)-333.2(4.1.)]TJ /F7 1 Tf 3.0465 -2.9132 TD [(5.)-575(Time)-383.2(complexit)32.1(y)-383.3(analysis)-383.3(and)-383.3(impro)32(v)32(emen)32(ts)]TJ /F2 1 Tf -3.0465 -2.3653 TD [(It)-298.1(is)-298(straigh)27.8(tforw)27.8(ard)-298(to)-298(calculate)-298(the)-298(time)-298.1(complexit)27.9(y)-298.1(o)0(f)-298.1(the)-298(algorithm.)-432.6(In)]TJ -2.0091 -1.2694 TD 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[(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 ()Tj /F2 1 Tf 10.9091 0 0 10.9091 90.5434 316.6924 Tm (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 108.8767 321.1128 Tm (21)Tj /F11 1 Tf 1.0625 0 TD (/)Tj /F1 1 Tf 0.5312 0 TD (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 128.1295 316.6924 Tm (n)Tj /F2 1 Tf 0.6003 0 TD (\))Tj /F10 1 Tf 0.3889 0 TD (.)Tj -3.8654 -2.1352 TD [(Pr)51.1(o)51.2(o)0(f)]TJ /F2 1 Tf 2.3267 0 TD 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[(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(eried)-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 ()Tj /F2 1 Tf 10.9091 0 0 10.9091 458.7338 701.071 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (r)Tj /F2 1 Tf 0.6457 0 TD (log)Tj /F1 1 Tf 7.9701 0 0 7.9701 484.1096 705.4915 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 490.66 701.071 Tm (n)Tj /F2 1 Tf 0.6003 0 TD [(\))-465.7(steps.)]TJ -30.6625 -1.2694 TD [(Th)27.8(us)-389.7(the)-389.7(time)-389.7(complexit)27.8(y)-389.7(o)0(f)-389.7(step)-389.7(5)-389.8(i)0(s)]TJ /F8 1 Tf 17.1622 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 358.5574 691.1821 Tm ()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 ()Tj /F2 1 Tf 10.9091 0 0 10.9091 178.4179 673.3749 Tm (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 196.7512 677.7953 Tm (21)Tj /F11 1 Tf 1.0625 0 TD (/)Tj /F1 1 Tf 0.5312 0 TD (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 216.004 673.3749 Tm (n)Tj /F2 1 Tf 0.6002 0 TD [(\).)-697.2(This)-417.5(time)-417.5(dominates)-417.5(all)-417.6(the)-417.6(others)-417.5(and)-417.5(is)-417.6(therefore)-417.4(the)-417.6(time)]TJ -5.4856 -1.2694 TD [(complexit)27.9(y)-333.3(o)0(f)-333.3(the)-333.4(algorithm.)]TJ ET 526.53 666.249 6.775 -0.398 re f 526.53 665.948 0.398 -5.978 re f 532.89 665.948 0.399 -5.978 re f 526.53 659.889 6.775 -0.398 re f BT 10.9091 0 0 10.9091 184.6266 635.0745 Tm [(The)-339.3(time)-339.3(complexit)27.9(y)-339.3(o)0(f)-339.4(the)-339.3(algorithm)-339.3(can)-339.3(b)-27.7(e)-339.4(impro)27.8(v)27.8(e)0(d)-339.3(b)27.8(y)-339.3(impro)27.9(ving)-339.4(the)]TJ -2.0091 -1.2694 TD [(estimate)-267.4(for)]TJ /F8 1 Tf 5.4045 0 TD (r)Tj /F2 1 Tf 0.7464 0 TD [(\(done)-267.5(in)-267.5(Lemma)-267.5(4.3\).)-422.5(Of)-267.4(course)-267.5(the)-267.5(b)-27.7(est)-267.5(p)-27.7(ossible)-267.4(scenario)-267.4(w)27.8(ould)]TJ -6.1509 -1.2694 TD [(b)-27.7(e)-348.3(when)]TJ /F8 1 Tf 4.0022 0 TD (r)Tj /F2 1 Tf 0.7817 0 TD (=)Tj /F8 1 Tf 1.0806 0 TD (O)Tj /F2 1 Tf 0.7905 0 TD (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 253.6429 611.7998 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 260.1933 607.3784 Tm (n)Tj /F2 1 Tf 0.6003 0 TD [(\))-348.4(and)-348.3(in)-348.3(that)-348.3(case)-348.3(the)-348.3(time)-348.3(complexit)27.9(y)-348.4(o)0(f)-348.3(the)-348.3(algorithm)]TJ -9.5364 -1.2694 TD [(w)27.8(ould)-498.2(b)-27.7(e)]TJ /F8 1 Tf 4.6075 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 221.597 597.4894 Tm ()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 ()Tj /F8 1 Tf 1.0556 0 TD (m)Tj /F2 1 Tf 1.1988 0 TD [(for)-320.8(whic)27.8(h)-320.9(o)]TJ /F11 1 Tf 7.9701 0 0 7.9701 391.5786 540.3876 Tm (q)Tj /F2 1 Tf 10.9091 0 0 10.9091 396.1435 542.0234 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD 0.2778 Tc (\)=)Tj /F8 1 Tf 1.7222 0 TD 0 Tc (q)Tj /F9 1 Tf 0.6796 0 TD ()Tj /F2 1 Tf 0.9751 0 TD [(1)-320.8(i)0(s)-320.9(asymptotically)]TJ /F8 1 Tf -25.7642 -1.2694 TD (A)Tj /F2 1 Tf 0.75 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD (\))Tj /F9 1 Tf 0.6111 0 TD ()Tj /F11 1 Tf 7.9701 0 0 7.9701 199.2319 532.4713 Tm (m)Tj ET 194.97 531.166 15.959 -0.436 re f BT /F1 1 Tf 7.9701 0 0 7.9701 194.9978 524.4135 Tm (ln)Tj /F11 1 Tf 1.0625 0 TD (m)Tj /F2 1 Tf 10.9091 0 0 10.9091 215.7888 528.1754 Tm (where)Tj /F8 1 Tf 2.8916 0 TD (A)Tj /F2 1 Tf 0.75 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD [(\))-333.3(i)0(s)-333.3(Artins)-333.2(constan)27.8(t)-333.3(with)]TJ /F8 1 Tf 11.4083 0 TD (A)Tj /F2 1 Tf 0.75 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD (\))Tj /F8 1 Tf 0.6667 0 TD (>)Tj /F2 1 Tf 1.0555 0 TD (0)Tj /F8 1 Tf 0.5 0 TD (.)Tj /F2 1 Tf 0.2778 0 TD (35.)Tj /F10 1 Tf -23.1348 -2.1827 TD (Sophie)Tj /F2 1 Tf 2.9374 0 TD (-)Tj /F10 1 Tf 0.3333 0 TD [(Germain)-530.3(Prime)-530.4(Density)-530.4(Conje)51.1(ctur)51.1(e)]TJ /F2 1 Tf 16.1033 0 TD [(.)-2015.2(The)-521(n)27.8(u)0(m)27.8(b)-27.8(er)-521(of)-521(primes)]TJ /F8 1 Tf -21.3831 -1.2694 TD (q)Tj /F9 1 Tf 0.9005 0 TD ()Tj /F8 1 Tf 1.196 0 TD (m)Tj /F2 1 Tf 1.2956 0 TD [(suc)27.8(h)-417.6(that)-417.6(2)]TJ /F8 1 Tf 5.0908 0 TD (q)Tj /F2 1 Tf 0.7607 0 TD [(+)-278.4(1)-417.6(is)-417.6(also)-417.6(a)-417.5(prime)-417.6(is)-417.5(asymptotically)]TJ /F1 1 Tf 7.9701 0 0 7.9701 449.0361 494.8951 Tm (2)Tj /F11 1 Tf 0.5312 0 TD (C)Tj /F17 1 Tf 4.9813 0 0 4.9813 459.3056 493.8989 Tm (2)Tj /F11 1 Tf 7.9701 0 0 7.9701 463.1941 494.8951 Tm (m)Tj ET 449.01 493.486 21.649 -0.436 re f BT /F1 1 Tf 7.9701 0 0 7.9701 449.9368 486.3073 Tm (ln)Tj /F17 1 Tf 4.9813 0 0 4.9813 456.9933 489.3798 Tm (2)Tj /F11 1 Tf 7.9701 0 0 7.9701 462.2934 486.3073 Tm (m)Tj /F2 1 Tf 10.9091 0 0 10.9091 476.4354 490.5166 Tm (where)Tj /F8 1 Tf 2.976 0 TD (C)Tj /F1 1 Tf 7.9701 0 0 7.9701 516.6974 488.8807 Tm (2)Tj /F2 1 Tf 10.9091 0 0 10.9091 525.9855 490.5166 Tm (is)Tj -33.3004 -1.5343 TD [(the)-333.8(t)27.8(win)-333.8(prime)-333.9(constan)27.9(t)-333.9(\(estimated)-333.7(to)-333.8(b)-27.8(e)-333.8(appro)27.9(ximately)-333.8(0)]TJ /F8 1 Tf 25.4654 0 TD (.)Tj /F2 1 Tf 0.2778 0 TD [(66\).)-446(Primes)]TJ /F8 1 Tf 5.4688 0 TD (q)Tj /F2 1 Tf 0.8162 0 TD (with)Tj -32.0281 -1.2694 TD [(this)-333.3(prop)-27.8(ert)27.8(y)-333.3(are)-333.3(called)-333.3(Sophie-Germain)-333.2(primes.)]TJ 2.0091 -2.1826 TD [(Artins)-222.2(conjectureif)-222.2(it)-222.2(b)-27.8(ecomes)-222.1(eectiv)27.8(e)-222.2(for)]TJ /F8 1 Tf 19.4639 0 TD (m)Tj /F2 1 Tf 1.1558 0 TD (=)Tj /F8 1 Tf 1.0555 0 TD (O)Tj /F2 1 Tf 0.7905 0 TD (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 448.0409 440.5419 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 454.5923 436.1206 Tm (n)Tj /F2 1 Tf 0.6002 0 TD (\)immediately)Tj -27.3562 -1.2694 TD [(sho)27.8(ws)-358.8(that)-358.8(there)-358.8(is)-358.8(an)]TJ /F8 1 Tf 10.1191 0 TD (r)Tj /F2 1 Tf 0.7992 0 TD (=)Tj /F8 1 Tf 1.0981 0 TD (O)Tj /F2 1 Tf 0.7905 0 TD (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 320.7532 426.6939 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 327.3036 422.2725 Tm (n)Tj /F2 1 Tf 0.6003 0 TD [(\))-358.8(with)-358.8(the)-358.8(required)-358.8(prop)-27.8(erties.)-520.8(There)-358.8(has)]TJ -15.6881 -1.2694 TD [(b)-27.7(een)-410.6(some)-410.5(progress)-410.5(to)27.8(w)27.8(ards)-410.5(pro)27.8(ving)-410.5(Artins)-410.5(conjecture)-410.4([GM],)-429.8([GMM],)-429.8([HB],)]TJ T* [(and)-282.7(it)-282.7(is)-282.8(also)-282.7(kno)27.8(wn)-282.8(that)-282.7(this)-282.7(conjecture)-282.7(holds)-282.7(under)-282.7(the)-282.8(Generalized)-282.6(Riemann)]TJ T* [(Hyp)-27.7(othesis.)]TJ 2.0091 -1.2694 TD [(If)-333.3(the)-333.3(second)-333.3(conjecture)-333.3(holds,)-333.2(w)27.8(e)-333.4(can)-333.3(conclude)-333.2(that)]TJ /F8 1 Tf 23.2361 0 TD (r)Tj /F2 1 Tf 0.7567 0 TD (=)Tj /F8 1 Tf 1.0556 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 466.5056 370.8394 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 473.5901 366.8802 Tm (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 491.9243 371.3016 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 498.4747 366.8802 Tm (n)Tj /F2 1 Tf 0.6002 0 TD 0.0001 Tc (\):)Tj -28.8787 -2.1795 TD 0 Tc [(By)-293.5(the)-293.5(densit)27.8(y)-293.5(o)0(f)-293.5(Sophie-Germain)-293.4(primes,)-301.4(there)-293.5(m)27.8(ust)-293.5(exist)-293.5(at)-293.6(least)]TJ 0 -1.2694 TD (log)Tj /F1 1 Tf 7.9701 0 0 7.9701 204.0727 333.6767 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 210.6232 329.2563 Tm (n)Tj /F2 1 Tf 1.0809 0 TD [(suc)27.8(h)-480.6(primes)-480.6(b)-27.8(e)0(t)27.8(w)27.8(een)-480.6(8)-166.7(log)]TJ /F1 1 Tf 7.9701 0 0 7.9701 350.5704 333.6767 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 357.1208 329.2563 Tm (n)Tj /F2 1 Tf 1.0809 0 TD (and)Tj /F8 1 Tf 2.0918 0 TD (c)Tj /F2 1 Tf 0.5994 0 TD (log)Tj /F1 1 Tf 7.9701 0 0 7.9701 412.3617 333.6767 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 418.9121 329.2563 Tm (n)Tj /F2 1 Tf 0.6003 0 TD [(\(log)-180.6(log)]TJ /F8 1 Tf 3.3055 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD (\))Tj /F1 1 Tf 7.9701 0 0 7.9701 472.3109 333.2155 Tm (2)Tj /F2 1 Tf 10.9091 0 0 10.9091 482.2874 329.2563 Tm [(for)-480.7(a)]TJ -26.7947 -1.2694 TD [(suitable)-461.4(constan)27.8(t)]TJ /F8 1 Tf 8.0174 0 TD (c)Tj /F2 1 Tf 0.4328 0 TD [(.)-828.9(F)83.4(or)-461.6(an)27.8(y)-461.4(suc)27.8(h)-461.5(prime)]TJ /F8 1 Tf 10.3943 0 TD (q)Tj /F2 1 Tf 0.4823 0 TD [(,)-493.5(either)-461.4(o)]TJ /F11 1 Tf 7.9701 0 0 7.9701 447.0267 313.7713 Tm (q)Tj /F2 1 Tf 10.9091 0 0 10.9091 451.5916 315.4082 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD (\))Tj /F9 1 Tf 0.8802 0 TD ()Tj /F2 1 Tf 1.2692 0 TD 0.4615 Tc [(2o)461.5(r)]TJ -27.1195 -1.2694 TD 0 Tc (o)Tj /F11 1 Tf 7.9701 0 0 7.9701 195.4361 299.9233 Tm (q)Tj /F2 1 Tf 10.9091 0 0 10.9091 200.001 301.5602 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6003 0 TD (\))Tj /F9 1 Tf 0.7611 0 TD ()Tj /F11 1 Tf 7.9701 0 0 7.9701 232.8349 306.4917 Tm (q)Tj /F12 1 Tf 0.5102 0 TD ()Tj /F1 1 Tf 0.8265 0 TD (1)Tj ET 232.89 304.486 14.887 -0.436 re f BT 7.9701 0 0 7.9701 238.1619 297.7983 Tm (2)Tj /F2 1 Tf 10.9091 0 0 10.9091 248.9185 301.5602 Tm 0.6145 Tc [(.A)614.5(n)642.4(y)]TJ /F8 1 Tf 3.0878 0 TD 0 Tc (q)Tj /F2 1 Tf 0.8723 0 TD [(for)-390(whic)27.8(h)-390(o)]TJ /F11 1 Tf 7.9701 0 0 7.9701 346.7198 299.9233 Tm (q)Tj /F2 1 Tf 10.9091 0 0 10.9091 351.2847 301.5602 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD (\))Tj /F9 1 Tf 0.7612 0 TD ()Tj /F2 1 Tf 1.15 0 TD [(2)-390(m)27.9(ust)-390(divide)]TJ /F8 1 Tf 6.4533 0 TD (n)Tj /F1 1 Tf 7.9701 0 0 7.9701 459.8715 305.5193 Tm (2)Tj /F9 1 Tf 10.9091 0 0 10.9091 467.4401 301.5602 Tm ()Tj /F2 1 Tf 1.0378 0 TD [(1)-390(and)]TJ -26.4715 -1.2694 TD [(so)-318.6(the)-318.5(n)27.7(u)0(m)27.8(b)-27.7(er)-318.6(of)-318.5(suc)27.8(h)]TJ /F8 1 Tf 9.9123 0 TD (q)Tj /F2 1 Tf 0.8009 0 TD [(is)-318.5(b)-27.8(ounded)-318.5(b)27.8(y)]TJ /F8 1 Tf 6.4335 0 TD (O)Tj /F2 1 Tf 0.7906 0 TD (\(log)Tj /F8 1 Tf 1.8472 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD [(\).)-439.5(This)-318.5(implies)-318.5(that)]TJ -20.3848 -1.2694 TD [(there)-348.5(m)27.8(ust)-348.6(exist)-348.5(a)-348.6(prime)]TJ /F8 1 Tf 11.1484 0 TD (r)Tj /F2 1 Tf 0.7821 0 TD (=)Tj /F8 1 Tf 1.0809 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 340.549 277.8231 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 347.6334 273.864 Tm (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 365.9666 278.2845 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 372.5171 273.864 Tm (n)Tj /F2 1 Tf 0.6003 0 TD [(\))-348.6(suc)27.8(h)-348.6(that)-348.5(o)]TJ /F11 1 Tf 7.9701 0 0 7.9701 441.1397 272.2272 Tm (r)Tj /F2 1 Tf 10.9091 0 0 10.9091 445.6947 273.864 Tm (\()Tj /F8 1 Tf 0.3889 0 TD (n)Tj /F2 1 Tf 0.6002 0 TD (\))Tj /F8 1 Tf 0.6921 0 TD (>)Tj /F2 1 Tf 1.081 0 TD (log)Tj /F1 1 Tf 7.9701 0 0 7.9701 489.9169 278.2845 Tm (2)Tj /F8 1 Tf 10.9091 0 0 10.9091 496.4683 273.864 Tm (n)Tj /F2 1 Tf 0.6002 0 TD (.)Tj -28.6948 -1.2694 TD [(Suc)27.8(h)-280.9(a)0(n)]TJ /F8 1 Tf 3.7005 0 TD (r)Tj /F2 1 Tf 0.7598 0 TD [(will)-280.8(yield)-280.8(an)-280.8(algorithm)-280.7(with)-280.8(time)-280.8(complexit)27.8(y)]TJ /F8 1 Tf 19.5243 0 TD (O)Tj /F12 1 Tf 7.9701 0 0 7.9701 460.2571 263.9751 Tm ()Tj /F2 1 Tf 10.9091 0 0 10.9091 467.3415 260.0159 Tm (\(log)Tj /F1 1 Tf 7.9701 0 0 7.9701 485.6747 264.4364 Tm (6)Tj /F8 1 Tf 10.9091 0 0 10.9091 492.2252 260.0159 Tm (n)Tj /F2 1 Tf 0.6003 0 TD (\).)Tj -28.7968 -2.1795 TD [(There)-241(has)-241(b)-27.7(een)-241(progress)-240.9(to)27.8(w)27.7(ards)-240.9(pro)27.8(ving)-241(this)-240.9(conjecture)-241(as)-241(w)27.8(ell.)-413.7(Let)]TJ /F8 1 Tf 29.5268 0 TD (P)Tj /F2 1 Tf 0.7809 0 TD (\()Tj /F8 1 Tf 0.3889 0 TD (m)Tj /F2 1 Tf 0.878 0 TD (\))Tj -33.5837 -1.2694 TD [(denote)-409.9(the)-410.1(greatest)-410(prime)-409.9(divisor)-410(of)-410(n)27.8(u)0(m)27.8(b)-27.8(er)]TJ /F8 1 Tf 20.1426 0 TD (m)Tj /F2 1 Tf 0.8781 0 TD 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