Pythagorean triples A triple of positive integers a, b, and...

Pythagorean triples A triple of positive integers $a, b,$ and $c$ is called a Pythagorean triple if $a^{2}+b^{2}=c^{2} .$ Let $a$ be an odd positive integer and let $$ b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \text { and } c=\left\lceil\frac{a^{2}}{2}\right\rceil $$ be, respectively, the integer floor and ceiling for $a^{2} / 2$ a. Show that $a^{2}+b^{2}=c^{2} .$ (Hint: Let $a=2 n+1$ and express $b$ and $c$ in terms of $n .$ b. By direct calculation, or by appealing to the accompanying figure, find $$ \lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{ | \frac{a^{2}}{2} \rceil} $$

Pythagorean triples A triple of positive integers $a, b,$ and $c$ is
called a Pythagorean triple if $a^{2}+b^{2}=c^{2} .$ Let $a$ be an odd
positive integer and let
$$
b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \text { and } c=\left\lceil\frac{a^{2}}{2}\right\rceil
$$
be, respectively, the integer floor and ceiling for $a^{2} / 2$
a. Show that $a^{2}+b^{2}=c^{2} .$ (Hint: Let $a=2 n+1$ and express
$b$ and $c$ in terms of $n .$
b. By direct calculation, or by appealing to the accompanying
figure, find
$$
\lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{ | \frac{a^{2}}{2} \rceil}
$$

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Key Concepts

Limits and Asymptotic Behavior

This concept involves studying the behavior of sequences or functions as the input grows without bound. It is used to understand long-term trends and approximate values, such as finding the limit of a ratio as the variable approaches infinity.

Pythagorean Triples

A set of three positive integers (a, b, c) that satisfy the equation a² + b² = c². This concept is widely used in number theory and geometry to explore integer solutions to quadratic equations and to construct triangles with integer side lengths.

Floor and Ceiling Functions

These functions map a real number to the nearest integers in specific directions: the floor function returns the greatest integer less than or equal to the number, while the ceiling function returns the smallest integer greater than or equal to the number. They are fundamental in discrete mathematics for handling approximations and discrete intervals.

Representation of Odd Integers

Expressing odd integers in the form 2n + 1 simplifies algebraic manipulation and proofs involving parity. This representation is particularly useful in number theory when establishing properties that hold specifically for odd numbers.