|
Pythagorean Triples
By Kelly Edenfield
The Pythagorean theorem is used in all aspects of mathematics. However, the algebra assoicated with the generating of the triples that satisfy the theorem is often overlooked. Just as the nature of the theorem was known before Pythagoras' time, so was the idea of the triples. As early as 1900 to 1600 B. C. the Babylonians demonstrated an awareness of at least 15 triples. The method by which they arrived at the triples was not reported on the cuneiform tablet, but the magnitude of the numbers (eg. 12,709, 13,500, and 18,541) indicates that trial and error could not have been the sole method of generating the triples. The Egyptians and writings from the Hindu Sulvasutras indicate the use of teh 3-4-5 triangle in constructions of pyramids and temples; however, neither group recorded a method of obtaining the sets of numbers.
Pythagoras (c. 540 B.C.) presented a formula for generating triples. By using the properties of figurate numbers, Pythagoras created a geometric representation of the square numbers. By creating a square array of dots, we can generate perfect squares. The addition of another row and column of dots yields another perfect square. This type of configuration is a "gnomon."
If we begin with a square figure representing n2, the dots placed around it will be 2n + 1, and the resulting square will be (n + 1)2. That is, n2 + (2n + 1) = (n + 1)2.
Since we are looking for triples of squared numbers, let 2n + 1 = m2. Then, n = (m2 - 1)/2 and n + 1 = (m2 + 1)/2. This yields
m2 + ((m2 - 1)/2)2 = ((m2 + 1)/2)2, where m must be odd.
Plato (c. 380 B. C.) is attributed with a second formula: (2m)2 + (m2 - 1)2 = (m2 + 1)2, where m is any natural number. This equation is derivable from Pythagoras' by multiplying the Greek formula by 4. Since it is more general, allowing m to be even or odd, Plato's formula is preferred to Pythagoras'.
A more general formula for obtaining all triples was given by Euclid (c. 300 B.C.) in his book Elements: (2uv)2 + (u2 - v2)2 = (u2 + v2)2, where u and v have no common factors, u > v, amd one of u and v is odd and the other is even. More complicated versions of this formula were also known to Diophantus of Alexandria and to Brahmagupta (c. 628) and Bhaskara (c. 1150) in India.
With the differing formulas, many questions arise: Are the formulas of Plato and Pythagoras essentially the same? That is, do they generate the same or different sets of triples? As stated above, by multiplying the Greek formula by 4, Plato's formula is derived. Would multiplying by 2 generate triples?
Multiplying by 2 gives the following formula:
(m2 + 1)2/2 = (m2 - 1)2/2 + 2m2
In the following spreadsheets, we can see that this new formula is definitely not an improvement over either of the two ancient formulas.
Click here to see Excel spreadsheets for the Pythagoras' formula, Plato's formula, and Pythagoras times 2.
In the spreadsheets, the numbers in columns B, C, and D are the squares of the triples.
After the most known triple (3-4-5), the two formulas do not seen to generate all of the primitive triples (triples with no common factors). It is possible, however, that the two work together to generate all primitive triples.
Pythagoras' formula allows only the use of odd numbers for m. As a result, this formula generates all primitive triples whose lowest terms are the odd numbers (greater than 1). That is, all triples with an odd number as the lowest a or b in the Pythagorean theorem can be generated from this formula. These are all primitive triples.
The first shortcoming of Plato's formula is that it will not produce triples whose longer side and hypontenuse differ by one. This failing will be accounted for later.
Apart from the 3-4-5 triple, the triples in Plato's formula are all different from those in the Pythagorean spreadsheet. In contrast to Pythagoras' formula, the m in Plato's formula can be any natural number. As a result, all triples with lowest term that is an even number greater than 2 are generated. However, this means that multiples of primitive triples are generated in addition to primitive triples. In examining the spreadsheet, we can see that when m is even, primitive triples are generated. When m is odd (and greater than 1), multiples of 2 of triples are generated.
Therefore, Plato's formula is superior to that of Pythagoras and the Greeks. To generate all primitive triples by Plato's formula, we need only divide the triples that have a common factor of 2 by 2. This formula and method will provide all of the Pythagorean triples.
A third formula mentioned above is Euclid's formula. To use this formula, u and v must be chosen such that one is even and one is odd. They are not as easily selected as the m's from the above formulas. Click here to see a spreadsheet of Euclid's formula.
|
Printer-friendly format
Email this thread to a friend
Bookmark this thread
(1000+ posts)
Send PM |
Profile |
Ignore
>
