Find in Library

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>b</mi></semantics></math></inline-formula> be an odd number. By using elementary methods, we prove that: (1) When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>x</mi></semantics></math></inline-formula> is an odd number and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>y</mi></semantics></math></inline-formula> is an even number, the Diophantine equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msup><mn>2</mn><mi>x</mi></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>b</mi><mi>y</mi></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> has no positive integer solution except when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>b</mi></semantics></math></inline-formula> is two special types of odd number. (2) When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>x</mi></semantics></math></inline-formula> is an odd number and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>≡</mo><mo>±</mo><mn>3</mn><mo stretchy="false">(</mo><mi>mod</mi><mn>8</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, the Diophantine equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msup><mn>2</mn><mi>x</mi></msup><mo>−</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>b</mi><mi>y</mi></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup><mi>z</mi><mn>2</mn></msup></mrow></semantics></math></inline-formula> has no positive integer solution except where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> and is another special type of the odd number.