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.
2 U; o+ w  k0 g6 y7 o% A2. Programme Rowland's formular and verify his results. Try different starting values and see what happens.
! T. `- y3 o# l8 E9 J& {6 @3 |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?
. ?/ L# C  S. F" Q4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.
2 g2 f# o. v, e3 [3 z! {$ U5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?
6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.
3 |" l+ y0 o+ G. }5 X* C7. Can pq be a Carmichael number where p and q are odd primes.
3 ^' y1 ]1 u: v; w8. 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.
% @6 z% R+ X4 J' C( Z% d9. Apply the Rabin-Miller test to n=1729 and n=2465
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.