求解题-数论基础(英文题目，信息安全研究生)-~~英文好的进

eternityran

1. Verify that Φ(84)=Φ(12)Φ(7) by finding a bijection between ordered pairs.
# r& b4 B/ R% v& k/ Y2. Programme Rowland's formular and verify his results. Try different starting values and see what happens.
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?
  A7 e" [/ P) V3 k- X4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.
. ~2 B9 ^8 Y2 K# J5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?
% Z# S( L& O  D/ N& d# ?6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.
7. Can pq be a Carmichael number where p and q are odd primes.
2 S5 H& `2 _$ u' ], z) W2 D8. 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.
3 B2 J" ~) Z8 I2 F* K: S4 T9. Apply the Rabin-Miller test to n=1729 and n=2465
( I$ X! f/ k) o10. 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.