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. 1 `, F! S# Q3 g+ |3 V2. Programme Rowland's formular and verify his results. Try different starting values and see what happens. + r1 {4 T: q( x) L* G2 \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? " K" g* P5 x" Z8 v" a6 a$ p4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.( L) m$ @0 M- Z6 g& U" @
5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13? 2 t$ Z2 J' b% y, a6 t5 J6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly. 5 i- }9 @1 ^% D/ o' t# f' }7. Can pq be a Carmichael number where p and q are odd primes. : A+ C0 r5 ]5 h* |; @, d* H6 C: ~ K8. 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.- }% n, w" y: P/ p+ {/ a( Y5 U
9. Apply the Rabin-Miller test to n=1729 and n=2465 % v9 c. J1 p5 s! d, F10. 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.