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.
! T2 t  ^7 G# k3 m; S6 Y8 Z2. 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?
- s8 i7 `  o8 p6 }) e. F9 m4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also.
' r; v5 F6 ]$ C; L2 Z/ a5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?
& `4 C. P. P0 Z9 W+ {: D& ]# y6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly.
) ?# N" l: v: H( P1 E4 D1 @9 n; n7. Can pq be a Carmichael number where p and q are odd primes.
0 X" d9 J0 e* q7 b( I2 t, I9 u; Y8. 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.
1 B0 T2 e1 T  i" w+ E7 ]9. 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.