Mathematical Induction

Prove that 2(2^n-1) = (n+1)/1 + ... + (n+1)/n for every natural number n.

Mathematical Induction

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Quadratic Reciprocity Law

Prove, using the quadratic reciprocity law that (3/p) = 1 if p== 1 or 11 (mod 12) (3/p) = -1 if p==5 or 7 (mod12) for each odd prime p.

Jacobi Symbols and Proofs

1. Prove that if b and c are odd, then (a/bc)=(a/b)(a/c) 2. Prove that if a==b (mod c), where c is odd, then (a/c)=(b/c)

Venn Diagram

Draw Venn diagrams for each side of the equation below, shading the appropriate region. Then give a careful proof using the element method... Please see attached for full question.