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., I( k6 {- q9 ?+ A5 G( y; E: E
2. Programme Rowland's formular and verify his results. Try different starting values and see what happens./ s* F/ d0 L8 _# I
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? 7 k7 {$ T6 C/ b4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also. * _% Q, U* k# k3 F. Z. M5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13? ) m7 D; W: u2 Z" r5 D4 w, i6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.* p8 G0 {3 H$ `4 v% T% w
7. Can pq be a Carmichael number where p and q are odd primes. ( t6 G" x1 }$ [+ J! G# x; P
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. 5 N; t& E: j7 \% |% }, T) p y) F$ ?8 u9. Apply the Rabin-Miller test to n=1729 and n=2465* D, ]5 ~2 x$ _# R
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.