The Pythagorean Triples
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We define the set denoted by the name Pythagorean Triples to be that set whose members comprise only those trios of positive integers (a, b, c) that satisfy the Pythagorean relation
From that definition we want to devise an algorithm or a formula that enables us, given a single number, to determine whether it is a member of a Pythagorean Triple and to calculate the other two members of its trio. We want to find a way to generate all the elements of the set of Pythagorean Triples and only the members of that set. Because the criterion that defines the Pythagorean Triples involves square numbers, we begin our search by dissecting a square number.
By definition, we associate a square number with a collection of counters that can be laid out in a square array, an array that has an equal number of rows and columns, each marked out by laying the requisite number of counters in a straight line. Given one such array, we can generate the next larger array, the square of the next higher number of the counting sequence, by adding new counters along the top row, in the upper right corner, and along the right column of the original square; that is, we generate the next square in the sequence of square numbers by adding to it an odd number 2n+1 to create
Thus, for example, we generate the square of six (36) by adding eleven (2x5 + 1) to the square of five (25). If we look closely at that procedure and recall that one is considered to be a square number, then we can see that we can generate the N-th square number in the sequence of square numbers by adding up the first N odd numbers in the counting sequence. Because the N-th odd integer can be represented as 2N - 1 = 1+2(N-1), we can devise our desired formulae for finding Pythagorean Triples by exploiting the above description of square numbers.
Pick some square number c2 and remove the last odd number that was added to create that number. In accordance with the description of square numbers given above, we know that what's left over is also a square number, b2, whose root is b = c - 1. We now claim that the odd number that we removed is also a square number, a2; that is,
Solving that equation for c and then substituting c = b + 1 gives us the equations
Equations 4 comprise the first Cordova Equations. We require that these equations have only integer solutions, so the square of "a" must be an odd number so that the division by two will leave a remainder of one half to cancel the one half in Equation 4a. Thus, given any odd number "a" we can calculate the other two members of the Pythagorean Triple to which it belongs. If, for example, we let a = 3, then we obtain the familiar trio (3, 4, 5).
If we remove the last two odd numbers that were added to b2 to generate c2 and then claim that they add up to a square number a2, then we have (along with c = b + 2)
which leads to
Equations 2.5 comprise the second Cordova Equations. We require that these equations have only integer solutions, so the square of "a" must be a multiple of four. Thus, for any even number "a" we can calculate the other two members of the Pythagorean Triple to which it belongs. If, for example, we let a = 6, then we obtain (6, 8, 10), which is just double the familiar triple calculated above.
In general, if we remove the last f odd numbers that were added to b2 to generate c2 and claim that they add up to a square number themselves, then we have (along with c = b + f)
which leads to
Equations 8 comprise the general (or f-th) Cordova Equations. We require that these equations have only integer solutions, so we require in turn that "a" be a multiple of "f". For any given value of "f", then, we can determine the values of "a" that go with it and then calculate the corresponding values of "b" and "c".
The algorithm that comprises Equations 8 and their associated criterion for the relation between "a" and "f" definitely generates only legitimate Pythagorean Triples, but it doesn't generate all of them. Consider the following triples drawn from Reference 1:
Those triples all satisfy Equations 8, but we would not generate them by the criterion that we apply with those equations; the criterion for the relation between "a" and "f" fails us by not being broad enough. What we want to do now is to devise a new criterion that is simple to apply and that causes Equations 8 to generate all of the Pythagorean Triples.
We can see clearly in Equation 8a that if "f" is an odd number, then a2/f must also be an odd number and if "f" is an even number, then a2/f must also be an even number. Looking deeper we see that a2 must be an integer multiple of "f"; that is, we must have a = gf for some integer "g". But if "f" is a square number, we could also have In the table above we see an example of that possibility with f = 9 and a = 33 = 11x3. Now we can look even deeper and note that if we decompose our chosen number "a" into its prime factors (e.g. a = klmn), then we can use any "f" that decomposes into some multiple of those factors or their squares (e.g. f = k2m) that is less than a2. In the table above we see an example in the set a = 20 = 5x2x2 and f = 8 = 2x2x2. In that example the division of a square involving four twos by three twos leaves one factor of two to make the quotient an even number, as required by equation 8a.
So now Equations 8 will generate all of the Pythagorean Triples and only the Pythagorean Triples if the selected numbers "a" and "f" obey the criterion that the set of the prime factors of "f" constitutes a subset of the prime factors of a2 and the criterion that the prime factors of a2 other than the prime factors of "f" contain at least one factor of two if "f" contains at least one factor of two and contains no factors of two if "f" is an odd number.
I have named Cordova Equations for Felix Cordova, a sheet-metal worker who makes, among other things, window frames. The technique that he uses to ensure that the frames are perfectly rectangular is based upon the Pythagorean Theorem, so he possesses an intimate familiarity with the Pythagorean Equation (Equation 1) and a large number of Pythagorean Triples. That familiarity enabled Cordova to use the method of scientific induction to obtain his equations. In essence that method consists of laying out the data and then staring at them until the mind's natural tendency to discern patterns (even where none exist) generates a hypothetical rule that can be used to generate more presumable examples of the data. Errors in what the rule generates can then be used to guide the mind in modifying the rule until the modified rule generates new examples of the data free of error.
In contrast, I used the method of deduction to obtain the Cordova Equations and the criteria for choosing proper values for "a" and "f". Simply put, I laid out self-evident axioms and then manipulated them to generate theorems in accordance with the rules of logic. The fundamental axiom in this case was the description of a square number as a sum of odd integers, which axiom enabled me to reason that one square number can be split into a smaller square number plus a sum of successive odd numbers. Thence I was able to work out the Cordova Equations and their associated criteria for "a" and "f" as theorems.
The contrast between Cordova's empirical-inductive method and the axiomatic-deductive method that I used is no illusion. Being a Rationalist, I am tempted to believe that Cordova's subconscious mind worked out the deduction that I made and then pushed the resulting equations into his consciousness. But Cordova describes his method of discovery (in Reference 1) as consisting of writing the triples that he knew in the form of the Pythagorean Equation (i.e. 32 + 42 = 52, 62 + 82 = 102, etc.) and then examining the resulting list until he discerned the pattern a2/f = b + c (though not in that form at first) for the case f = 2, and then for f = 3 and f = 4. That method invalidates my suspicion because it betrays a lack of knowledge, conscious or subconscious, that square numbers are the sums of successive odd numbers. On the other hand, an empiricist, noting that the subconscious mind is conventionally regarded as irrational, would suspect that I gained the Cordova Equations via intuition, as Cordova did, though on different data, and then worked out the underlying algebra a posteriori, again intuitively. But I certainly could not have followed Cordova's path, because at the time I discovered the Cordova Equations for myself I knew only one Pythagorean Triple, the classic (3, 4, 5). However, as I worked on a different problem, the Babbage representation of the powers of integers, I reacquainted myself with the square numbers as sums of the odd integers. It was then easy to reason out that ablating one square number would leave another square number and a sum of odd numbers that might comprise yet another square number. A little simple algebra yielded the equation that describes that other square number.
While interesting in itself, that contrast may also help us to resolve the question of whether mathematics is invented or discovered. The fact that two different paths of discovery lead to the same conclusion strongly implies that the rules of mathematics are independent of human thought. Though we are certainly free to define and to name mathematical objects arbitrarily (e.g. square numbers), we are not free at all to decide what the relationships among those objects will be: those relationships are fixed and unalterable.
That unalterability of the relationships is a feature of Reality and is not merely a limitation of the human mind. We can demonstrate the truth of that proposition via a simple imaginary experiment. In a large field we assemble square tiles into square floors, one floor for each of the square numbers; that is, we have floors with 1, 4, 9, 16, 25, etc. tiles in them. To this field we bring an experimenter, a person who has no knowledge of mathematics but who has a perfect memory and who has a lot of time on his hands. We show him how to take tiles from one floor, lay them along the top row and the right side of another floor, and then reassemble them into their original floor. We then ask him to keep track of which floors, when combined in this manner, produce another perfectly square floor with no tiles missing or left over. When we return some time later our experimenter reports his results by pointing: this floor and this floor, he tells us (pointing to what we recognize as floors #3 and #4), make a square floor that's the same as this one (pointing to floor #5) and this floor and this floor (he points to floors #6 and #8) make a square floor that's the same as this one (he points to floor #10) and this floor and this floor.... Thus, a person with no knowledge of mathematics can, through trial and error, discover the Pythagorean Triples (actually a subset of them) by manipulating physical objects and relying on symmetry in their patterns to judge members of the required set. There's no room for our experimenter to invent anything: Reality constrains what he discovers.
Finally, we might note that different cultures, such as those of Greece, India, and China, even before they had discovered each other, had discovered the Pythagorean Theorem. That fact also argues for the proposition that mathematics is discovered, not invented. Whenever people anywhere begin thinking about mathematics, it seems, they will use different symbols and different names for the objects of their study, but they will all discover the same rules governing the relationships among those objects.
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