Suppose a and b are numbers that are relatively prime to p. Show that at least...
Please prove the 3 theorems,
thank you!
7.6 Theorem. Let p be a prime. Then half the numbers not congruent to 0 modulo p in any complete nesidue system modulo p are quadratic residuess modulo p and half are quadratic non-residues modulo p. From clementary school days, we have known that the product of a pos- itive number and a positive number is positive, a positive times a negative is negative, and the product of two negative numbers is positive....
Hello,
Can someone please show examples on how this proposition is
being used ( please make sure the example include i, ii, iii ?
Please be legible. Thank you.
COROLLARY 11.2.6 Ltp2 be prime, and let a, b e Z, with a, b + 0. Then: (i) lf-andb are both quadratic residues, then so is ab. ii) If a and b are both quadratic nonresidues, then ab is a quadratic residue. (ii) If one of a and b is a...
Exercise 5.6. Suppose a,b E Zt are show that am and 67" are relatively prime. If m and n are any positive integers, again relatively prime
Let p be a prime with p ≥ 13. Prove that among the integers 2,11 and 22, either all three are quadratic residues modulo p or exactly one is a quadratic residue modulo p.
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai?
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
problem 12
lengths are multiples of 3. What's up with that! 12. Suppose a and b are relatively prime. How does the length of the decimal expansion of compare to the lengths of the decimal expansions of and ? problems 11 and 12 ab
lengths are multiples of 3. What's up with that! 12. Suppose a and b are relatively prime. How does the length of the decimal expansion of compare to the lengths of the decimal expansions of and...
(1) The Legendre symbol and Euler's criterion. (1 pt each) Let p be an odd prime and a Z an integer which is not divisible by p. The integer a is called a quadratic residue modulo p if there is b E Z such that a b2 (p), i.e., if a has a square root modulo p. Otherwise a is called a quadratic non-residue. One defines the Legendr symbol as follows: 1 p)=T-i if a is a quadratic residue modulo...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
please prove proofs and do
7.4
7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime. han C: oadl Prove that if p, q, and r are three different prime numbers, then p2 + q2 #r2 11.