questions

Proof Methods: Deduction &amp; Exhaustion

The integers <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>p and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>q are both odd.
Prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">p^2+q^2</annotation></semantics></math>p2+q2 is two more than a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math>4.

Writes <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mi>r</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p=2r+1</annotation></semantics></math>p=2r+1 and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>=</mo><mn>2</mn><mi>s</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">q=2s+1</annotation></semantics></math>q=2s+1 for integers <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>r and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>s.

Expands and simplifies to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo stretchy="false">(</mo><msup><mi>r</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mo>+</mo><msup><mi>s</mi><mn>2</mn></msup><mo>+</mo><mi>s</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p^2+q^2=4(r^2+r+s^2+s)+2</annotation></semantics></math>p2+q2=4(r2+r+s2+s)+2.

Concludes from the form <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">4k+2</annotation></semantics></math>4k+2 that <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">p^2+q^2</annotation></semantics></math>p2+q2 is two more than a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math>4.