1. Verify that Φ(84)=Φ(12)Φ(7) by finding a bijection between ordered pairs.. g7 _. U/ ?! w' H
2. Programme Rowland's formular and verify his results. Try different starting values and see what happens.* b2 q8 _/ F B% c* y
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? ! [; \/ a- y; `4. Prove that if n is a pseudoprime to base 2 then 2^n-1 is a pseudoprime to base 2 also., H: D# j! ]5 D) q8 E
5. IS 341 a pseudoprime to the base 5? Is 341 a pseudoprime to base 7? Is 341 a pseudoprime to base 13?* u0 v. I* R% P, Q( T0 L% h# E4 Q& n
6. Verify that 1729 and 2465 are Carmichael numbers using the Korselt criterion and directly. 1 h4 c* s6 B/ A+ O8 w7. Can pq be a Carmichael number where p and q are odd primes. 9 X7 Z& m. \9 d# N8. 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.5 U0 V8 U4 d& Z$ f0 c8 ^1 F
9. Apply the Rabin-Miller test to n=1729 and n=2465/ |+ S. E4 g( p. M$ t
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.