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Raji 5.2, Primitive roots for primes: 8. Let r be a primitive root of p with p a primitive root. 1 (mod 4). Show that r is also I suppose p is a prime. Indeed, 2 is a primitive root modulo 9, but not. 2 is Write p = 4m + 1. As r is a primitive root, the numbers r; r2 ; r3 ; : : : ; r4m are a complete set of nonzero residues modulo p. Note that r2m 6= 1, but 2 r2m = 1, so r2m = 1 because p is a prime. Let s = r and consider the number s; s2 ; s3 ; : : : ; s4m . Note that if i is even, then si = ri while if i is odd, then si = ri . Suppose si = 1 for some i such that 1 i 4m. If i is even, then ri = 1 so i = 4m. If i is odd, then ri = 1. But r2m = 1, so i cannot be odd. So the only i with 1 i 4m such that si = 1 is i = 4m, which means that s is a primitive root. 9. Show that if p is a prime and p such that x2 1 (mod p). 1 (mod 4), then there is an integer x Didn’t we prove that in 8? The number x is rm . Raji 5.3, The existence of primitive roots: 1. Which of the integers 4, 12, 28, 36, 125 have primitive roots? Which are 2, 4, pe , or 2pe for an odd prime p? Only 4 and 125, right? 2. Find primitive roots of 4, 25, 18. For 4, the primitive root is 3. For 25, I would …rst try 2. The powers of 2 are 2; 4; 8; 16; 7; 14; 3; 6; 12; 24 = 1, so 210 1 and ord25 2 = 20 = ' (25). For 18, the …rst candidate is 5, whose powers are 5; 7; 1; 5; 7; 1. As ' (18) = ' (2) ' (9) = 6, we see that 5 is a primitive root of 18. 3. Find all primitive roots modulo 25. We know that 2 is a primitive root. The others are 2i where i is relatively prime to ' (25) = 20. So the primitive roots are 2, 23 , 27 , 29 , 211 , 213 , 217 , and 219 . Eight of them? Sure, because ' (20) = ' (4) ' (5) = 2 4 = 8. If you don’t want to write them as powers of 2, you can read them o¤ from the powers of 2 in the last exercise: 2, 8, 3, 12, 2, 8, 3, 12. Compare to Exercise 8 of 5.2 above. Raji 5.4, Introduction to quadratic residues and nonresidues: 1. Find all quadratic residues of 3. The nonzero squares modulo 3 are 12 = 1 and 22 = 4 quadratic residue of 3. 1 1, so 1 is the only 2. Find all quadratic residues of 13. Again, we simply list the square 12 = 1, 22 = 4, 32 = 9, 42 = 3, 52 = 12, 62 = 10. We can quit here because 7 = 6, 8 = 5, etc. So the quadratic residues are 1, 3, 4, 9, 10, and 12. Half of the twelve nonzero residues modulo 13. 3. Find all quadratic residues of 18. 2 2 12 = 1, 52 = 7, 72 = 13, 112 = ( 7) = 13, 132 = ( 5) = 7, 172 = 2 ( 1) = 1. So there are three of them: 1, 7, and 13. 2