标题: 求解题-数论基础(英文题目，信息安全研究生)-~~英文好的进 [打印本页] 作者: eternityran 时间: 2011-10-17 08:23 标题: 求解题-数论基础(英文题目，信息安全研究生)-~~英文好的进 1. Verify that Φ(84)=Φ(12)Φ(7) by finding a bijection between ordered pairs. ( @0 @; a9 c7 z2. Programme Rowland's formular and verify his results. Try different starting values and see what happens. |8 ?. {( b3 V$ h7 I3 x8 ]
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? ; D" P1 o- F4 M7 Q, }' _( F7 [4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also. 5 p, T/ n+ \' n5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?* b1 C7 J9 C) J
6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly. : }" d" ~: x3 `- d& B7. Can pq be a Carmichael number where p and q are odd primes. ' U4 { J$ G+ _8 \- v; C" e( [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.# P2 S) Q) y! k, U% n
9. Apply the Rabin-Miller test to n=1729 and n=2465 9 [. L9 C4 x0 h7 V/ ~( Z. Z) C10. 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.