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This investigation looks at the results when two square numbers are added together.
Here is a list of the first 11 prime numbers.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
(a) In the list there are 4 numbers that are one more than a multiple of 4.
These are called Pythagorean Primes.
The smallest one is 5 and the largest one is 29.
Write down the other two.
5, ..........., ..........., 29
(b) The 17th century French mathematician Albert Girard proved that every Pythagorean Prime equals the sum of two square numbers.
Write your answers to part (a) as the sum of two square numbers.
Two have been written down for you.
5 = 1^2 + 2^2
............ = ............ + ............
............ = ............ + ............
29 = 2^2 + 5^2
(c) Another Pythagorean Prime is 101.
Write 101 as the sum of two square numbers.
101 = ............ + ............
569
582
510
The sum of two square numbers can equal another square number. For example,
$$3^2 + 4^2 = 9 + 16 = 25 = 5^2$$
We say that $3, 4, 5$ is a \textit{Pythagorean Triple}.
(a) Show, by calculation, that $7, 24, 25$ is a Pythagorean Triple.
(b) Each row in this table is a Pythagorean Triple.
Complete the table. Use patterns of numbers in the table to help you.
[Table_1]
(c) What is the connection between the \textbf{square} of the smallest number and the other two numbers in each Pythagorean Triple in the table?
.....................................................................................................................................................
.................................................................................................................................................
(d) Use your answer to \textbf{part (c)} and the patterns of numbers in the table to complete the following Pythagorean Triple.
............ , ............ , 421
571
547
589
2\sqrt{x},\ x-1,\ x+1\text{ is a Pythagorean Triple when }x\text{ is a square number.}
(a) (i) Find the Pythagorean Triple when $x=16$.
\text{............ , ............ , ............}
(ii) Check that your answer to \text{part (a)(i) }\text{is a Pythagorean Triple.}
\text{Use the method of the example in \textbf{question 2}.}
X(b) In the table, $x$ is the square of an even number.
\text{Each row is a Pythagorean Triple.}
\begin{array}{|c|c|c|}\hline2\sqrt{x}&x-1&x+1\\\hline(x=16)&&\\\hline(x=36)&12&37\\\hline&16&63\\\hline&99&\\\hline24&&145\\\hline\end{array}
\text{Write your answer to \textbf{part (a)(i)} in the first row of this table.}
\text{Complete the three columns of the table.}
\text{You may use patterns or the fact that }2\sqrt{x},\ x-1,\ x+1\text{ is a Pythagorean Triple to help you.}
(c) \text{What is the connection between the \textbf{square} of the smallest number and the sum of the other two numbers in each of the Pythagorean Triples in the table?}
\text{.........................................................................................................................................}
\text{.........................................................................................................................................}
(d) \text{Show algebraically that }2\sqrt{x},\ x-1,\ x+1\text{ satisfies your connection in \textbf{part (c)}}.
555
538
530
