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?
, H: f: v' P$ r6 q+ t0 Y8 O4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.
, w& r" M1 ?4 ]+ H0 u+ p0 r0 h5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?
% Z& j; v; w/ J6 l6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.
9 w3 J- G: i' W& g# S7. Can pq be a Carmichael number where p and q are odd primes.
/ {' d( j1 t" v* ~" J2 v. E0 p8. 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 `, }. \" s. V9. Apply the Rabin-Miller test to n=1729 and n=2465
! c: D$ E  l' }! N- Y& \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.