标题: 求解题-数论基础(英文题目，信息安全研究生)-~~英文好的进 [打印本页] 作者: eternityran 时间: 2011-10-17 08:23 标题: 求解题-数论基础(英文题目，信息安全研究生)-~~英文好的进 1. Verify that Φ(84)=Φ(12)Φ(7) by finding a bijection between ordered pairs. 3 c* o9 v4 N ~8 t2 e# c2. Programme Rowland's formular and verify his results. Try different starting values and see what happens.+ h$ L" J6 l' J- X5 U! G
3. Verify the following result called Wilson's Theory: An integer n is prime if and only if (n-1)!≡-1(mod n ) for the cases n=2,3,4,5...,10. Can this be used as an efficent test for a prime? 4 W# J8 N/ @" N% [$ D4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.$ }9 O2 g: e& w0 I
5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13? 0 V, o9 G$ u/ ]. w0 ]3 `6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.6 W2 f; ^* ]* E! W; o# y
7. Can pq be a Carmichael number where p and q are odd primes. 9 u6 O5 d! @% s, b, J6 G- m
8. Find a k such that 6k+1,12k+1,18k+1 are all prime numbers. Prove that then n=(6k+1)(12k+1)(18k+1) is a Carmichael number.6 L; Y% I# H8 ~2 a
9. Apply the Rabin-Miller test to n=1729 and n=2465 2 p4 E* v* u; ?10. Let n=667. For which a is a^667≡a(mod 667). Do the same for n=833. You might need to write a ** programme in Maple.