MATH 261 Final, Fall 1996

LIBRA R y OF brine American University of Beirut Feb. 4, 1997 Math 261 Final Examination (Time: 2.5 hours) Prof. N. Nahlus 1. If mis square free and kj = 1mod 6(m), show that a°=a modm for all integers a, and why this is important for coding theory ? 2. Find all primitive solutions in positive integers of the equation x'+7y’=z' if y is even. 3. If p is an odd prime and (abc, p) = 1, show that the number of solutions (x, y) modulo p of the equation ax’ + by’ = cmod pis p - (=?) P ytk P > Hint: >) ( y=-l if & pel. y= n+2 4. Find the number of integers n, 1<n <p such that (2) = ( }=-l Pp P 5. If p isa prime, p =1 mod 4, and p=a’ +b’ witha odd and positive, show that (S) = 1. (Hint: Use the reciprocity law of the Jacobi symbol) P solvable if x” =a mod p, is solvable for each i. ( p,,p,,..., p, are distinct primes) 7. Compute 3) by Gauss Lemmas 1 and 2. LIBRARY 8. Find all integer solutions of 10x+ 14y + 35z = 1. or piney 9. Show that there are infinitely many primes of the form 8k+1 . Hint: Every prime divisor of x‘ + 1 is of the form 8k+1. (prove it !) 10. Let p and q be odd primes such that q = 2p+1 and p =1 mod 4. Show that 2 is a primitive root modulo q. Hint: Find 2) from p=4k+1. Then apply Euler’s Criterion. q 11. Let p be an odd prime and (a, p) = 1. Show that x"=a mod pis solvable if o(P") a “4 =1mod p” where d = g.c. d(n, o(p”)) Hint: Let g be a primitive root modulo p", x=g' and a= g”. Then solve for t. 12. Show that the number e is irrational. If e= 7 multiply both sides 13. Prove the reciprocity law of the Legendre symbol using Gauss sums knowing the fact concerning G’. ean