Quadratic congruences

8. Find all solutions to the quadratic congruences, if they exist. (a) x2 + x + 1 ≡ 0 (mod 7). (b) x2 ≡ 55 (mod 179)

Primitive numbers

Assume that n is odd and a is a primitive root mod n. Let b be an integer with b ≡ a(mod n) and gcd (b, 2n) =1. Show that b is a primitive root mod 2n.

Primitive root

Find a primitive root modulo 17 if it exists.

Primitive roots

Determine which elements of Z_7 (Z sub 7) are primitive roots.

Theory of Numbers (I): Principle of Mathematical Induction: Prove that 1^2 + 2^2 + 3^2 + … +n^2 = n(n+1)(2n + 1) / 6

Theory of Numbers (I) Principle of Mathematical Induction Prove that 1^2 + 2^2 + 3^2 + … +n^2 = n(n+1)(2n + 1) / 6

Theory of Numbers (II): Principle of Mathematical Induction: Prove that 1^3 + 2^3 + 3^3 + … + n^3 = (1 + 2 + 3 + … + n)^2

Theory of Numbers (II) Principle of Mathematical Induction Prove that 1^3 + 2^3 + 3^3 + … + n^3 = (1 + 2 + 3 + … + n)^2

Theory of Numbers (III): Principle of Mathematical Induction: Prove that x^n – y^n = (x – y)[ x^(n-1) + x^(n-2)y + … + xy^(n-2) + y^(n-1) ]

Theory of Numbers (III) Principle of Mathematical Induction Prove that x^n – y^n = (x – y)[ x^(n-1) + x^(n-2)y + … + xy^(n-2) + y^(n-1) ]

Theory of Numbers (IV): Principle of Mathematical Induction: Prove that 1.2 + 2.3 + 3.4 + … + n( n + 1) = n( n + 1)( n + 2 )/3

Theory of Numbers (IV) Principle of Mathematical Induction Prove that 1.2 + 2.3 + 3.4 + … + n( n + 1) = n( n + 1)( n + 2 )/3

Theory of Numbers (V): Principle of Mathematical Induction: Prove that 1 + 3 + 5 + …+(2n – 1) = n^2.

Theory of Numbers (V) Principle of Mathematical Induction Prove that 1 + 3 + 5 + …+(2n – 1) = n^2

Theory of Numbers (VI): Principle of Mathematical Induction: Prove that 1/(1.2) + 1/(2.3) + 1/(3.4) + … + 1/{n.(n + 1)} = n/(n + 1).

Theory of Numbers (VI) Principle of Mathematical Induction Prove that 1/(1.2) + 1/(2.3) + 1/(3.4) + … + 1/{n.(n + 1)} = n/ ...continues