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. $ F, s1 ` M4 |$ v) h- C, J2. Programme Rowland's formular and verify his results. Try different starting values and see what happens., t" H# F" g9 o4 V. z% I: W& f M
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? 6 R6 ^9 F* m- h+ I$ Y8 W# T4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also. 9 V6 t8 g2 u# }& G( }" _" I. g5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13? 5 O' S3 i5 U" E8 l5 Q6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.0 j9 R. D3 g8 s) _7 D
7. Can pq be a Carmichael number where p and q are odd primes. 5 B, N, `( B0 ~( K) T8. 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 y, g+ ^" x" N, B9. Apply the Rabin-Miller test to n=1729 and n=2465 8 j7 q& t$ V4 g0 W5 d10. 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.