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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 \quad 3 \quad 5 \quad 7 \quad 11 \quad 13 \quad 17 \quad 19 \quad 23 \quad 29 \quad 31
(a) In the list there are 4 numbers that are one more than a multiple of 4.
These are called \textit{Pythagorean Primes}.
The smallest one is 5 and the largest one is 29.
Write down the other two.
5, \text{..........., ...........,} 29
(b) The 17\textsuperscript{th} century French mathematician Albert Girard proved that every Pythagorean Prime equals the sum of two square numbers.
Write your answers to \textit{part (a)} as the sum of two square numbers.
Two have been written down for you.
5 \quad = \quad 1^2 \quad + \quad 2^2
\text{...........} \quad = \quad \text{..........} \quad + \quad \text{..........}
\text{...........} \quad = \quad \text{..........} \quad + \quad \text{..........}
29 \quad = \quad 2^2 \quad + \quad 5^2
(c) Another Pythagorean Prime is 101.
Write 101 as the sum of two square numbers.
101 \quad = \quad \text{..........} \quad + \quad \text{..........}
527
586
514
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 $Pythagorean\ Triple$.
(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]
3 4 5
5 12 13
7 24 25
9 40
11 60
13 113
(c) What is the connection between the $square$ of the smallest number and the other two numbers in each Pythagorean Triple in the table?
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(d) Use your answer to $part\ (c)$ and the patterns of numbers in the table to complete the following Pythagorean Triple.
............ , ............ , 421
514
559
587
2\sqrt{x}, \ x - 1, \ x + 1 \text{ is a Pythagorean Triple when } x \text{ is a square number.}
(a) \ (i) \ \text{Find the Pythagorean Triple when } x = 16.
......... , ......... , .........
(ii) \ \text{Check that your answer to part (a)(i) is a Pythagorean Triple.}
\text{Use the method of the example in question 2.}
(b) \ \text{In the table, } x \text{ is the square of an even number.}
\text{Each row is a Pythagorean Triple.}
$$\begin{array}{|c|c|c|}\hline\ 2\sqrt{x} & x - 1 & x + 1 \\ \hline\ (x = 16) && \\ \hline\ (x = 36) & 12 & 37 \\ \hline\ 16 & 63 & 65 \\ \hline\ 99 && \\ \hline\ 24 & & 145 \\ \hline\ \end{array}$$
\text{Write your answer to 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 square of the smallest number and the sum of the other two numbers in each of the Pythagorean Triples in the table?}
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(d) \ \text{Show algebraically that } 2\sqrt{x}, \ x - 1, \ x + 1 \text{ satisfies your connection in part (c).}
552
594
555
