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. ?- g" l) b4 E7 \% s
2. Programme Rowland's formular and verify his results. Try different starting values and see what happens.3 J, E& z2 `2 U" l- J
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?: |& u) T" L; h, t c
4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also. # D4 j0 {5 j0 z4 a T! n5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?/ S8 @+ e& b" P; A8 ]) V6 \- k9 h
6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.- d, m4 ?2 E S( N) m7 f9 W% X. p8 F
7. Can pq be a Carmichael number where p and q are odd primes. 7 q- a' v9 I! f6 O8. 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. 8 S4 V! \) B5 ~' F& M4 X9. Apply the Rabin-Miller test to n=1729 and n=2465 7 u; `: N- n7 l. v% H2 }3 l10. 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.