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. 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?
+ R  p' C! H8 P1 L# ^# f$ @4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.
" i8 U. w* f8 d- {5 k# `5. 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.
1 {' B7 L$ g8 H( J! S7. Can pq be a Carmichael number where p and q are odd primes.
" d* ~, p9 {2 N! L8. 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.
. b, ~$ n/ |5 j) r1 y9 B' X9 Y9. Apply the Rabin-Miller test to n=1729 and n=2465
/ w5 r4 e' }' w$ b, N. a* u10. 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.