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?
. L% O- d9 T# m. U0 y5 G/ Z4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.
5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?
: x6 ~7 F# }! h6 }7 s, `6 t# y9 H6. 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.
8. 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.
. e9 p! _5 D, h- F: o! O3 b9. Apply the Rabin-Miller test to n=1729 and n=2465
4 z5 v9 A1 x1 b. ?5 h1 H10. 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.