Question

please prove 10.1-10.2

LULUS MUTUCI U U esteu. 10.1 Exercise. If n is a d-digit number, explain why the trial division primality test requires roughly 104/2 trials. 10.2 Exercise. Ifn is a d-digit number, explain why the Wilson's Theorem primality test requires roughly 10% multiplications, These two algorithms are said to run in exponential time since the re-

Answer 1

2 - if n is a d-digit no. explain why the trial division primality test requires roughly 10 10d/ trials . gutast at will SOL:- positive integere, (The Prime No. Theorem) :- If n is a large then the number an of' primes Snºb approximately Ynn. More f :3 Gobb 14 06 616 de dobb bolo to bolo Theorem: Let p>1 be an hones integer. Then p has no prime drusor less than or equal to rp Lay på prime Proof: - Say p has no prime diuisor less than or equal to up. claim: pe prime. 2x Suppose not. po say, p is composite, and each prime dluisor p of p satisfies pi>vp på composite, p=pipe... PK is a product at eraction works of at least 2 primes. But then, igitur p = pipe... Pk > Wo so p3-.. Pk = pp3.. Preup. which shows psp which is a contradiction. conversely, say p is prime. Then the only prime dinsor of pop itself. Indeed, p> op so sincepsi. :p has no pane diuusor less than or equal to sp. conclusion: To test primality of o, we only need to do Trial divisions by primes & prif any one of these primes divide an is composite. Otherwise, as primer

say nhas d-digits . Then na iod. Using Prime No. Theorem In n. C-constant e mir Inn was on nalodas aa roa a tor zog (vn) e con {e= constant) sie te large du< 10d/2. . We have to check diuisibility for less than 10d/ 10.2) If n-6 d-digited, show Wilson's Theorem primality test requires roughly lod multiplications. H y Sol: (Wilson's Theorem) : - An Integer p> 16 prime qu<q bas (P-1) 1 = -1 (mode) So, to check if p s prime, we have to compute(p-1) So we have to multiply p-1 numbers. Also, p lod sering No.of steps regd is less than 10+ multiplicating

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Sane n has d-digits Than n10d eina Prime No.Theorem d In Vn . Ce onstant) Smivs In Vn Bn<j04 mists * 14u2 ess than lo/ . We have to check diui ited, show olson's Theosem n - test res nteger p1prime Sol:(Wason's Theorem tn (P-!-mod P ime malfiply p- nuumbers omputelp-)! So, to chack we have to So we hae to PKto Also, No. of sheps Aep less than 10 multpilcatiog 24N