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.7 B4 u& |' B5 b& G7 M
2. Programme Rowland's formular and verify his results. Try different starting values and see what happens. / H# m% D6 `0 G7 H3 h0 E8 J- x3. 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?# Y1 e H5 ^& ~) x; U
4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.. u" n* p+ W/ e% { I
5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?, {( h8 l: M v; T3 l. L2 A
6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.& y' w( Z% q' G r1 V, E
7. Can pq be a Carmichael number where p and q are odd primes. & d; A _6 D6 U( e/ i( F8. 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. % [9 e6 q/ s$ _, Z7 n; x3 {$ a9. Apply the Rabin-Miller test to n=1729 and n=2465! L+ h3 ?7 R. D* E
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.