Number Theory — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

5. (a) Prove that the quadratic congruence x'+5x+1=0 (mod p) has a solution for evers rime p, even though the equatior 6x'+5x+1= 0 (mod p) has no solution in the integers. 7.5 (b) (i) Primes oft it formate + initely many primes of the form 4k + 1. (ii) Show that 3 is quadratic residue of 23 but quadratic non residue of 31. (c) The cipher text VKYAQ VAKEC has been enciphered with the Linear Cipher Derive the plaintext. 7.5 6. (a) Prove that 2 is not a primitive root of any prime of the form p=3.2" +1 except when p (b) Find' the value of Legendre symbols (461/ (c) Use the Hill's cipher ClESPI+2P2(mod26) C2= 3P1 +4P2(mod 26) to encrypt the message GIVE THEM TIME. 1. (a) (i) Use the Euclidean Algorithm to find integers x and y satisfying gcd(1769, 2378) = 1769x + 2378y 4 (ii) Determine all solutions in the integers i of the Diophantine equation (b). Verify that O; 1, 2, 24, 2,......,2' form a complete set of residues modulo 11, but that O, i2, 2ª, 32, ..., 102 do not. (c) Obtain thé two incongruent solutions modulo 210 of the system 7.5 2. (a) (i) Make use of Fermat's theorem to prove that, if p is an odd. prime, then (it) For any integer a, verify that a, and 'a' have the same units digit. (b) If pis a prime, prove that for any integer plaP + (p-1)lạ and p|(p-1)laP ta (c) Prove that if n=p* pk.....*. is a prime factorization of n > 1, then 7.5 :: 3, (a), If I is, a, multiplicative, function and " i"' Then show that f is also multiplicative. (b) (i) For n › 2, Show that (n) is an even integer. ily Determine the day of 'the week (c) If F=2*" +1,n'>1 is a prime then show that 2 is not a primitive root of Fn* 7.5 4. (a) If the integer a has order k modulo n, then .:a'=al (mod n) if and only if isj (mod t). (b) (i) Determine all primitive roots of 11. " (ii) Ușe Euler Theorem to show that for any integer a, (c) For each positive integer n show that Also show that for any integer n23, Eu(k) =1