Basic Examples (2) 

Given a primitive Pythagorean triple, find its position on the Barning-Hall tree:

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See the tree:

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Scope (2) 

Use "PathWithFibonacciPythagoreanPair" to show the tree withFibonacci matrices from the root node to the target:

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Use "PathWithPythagoreanTriple" to show the tree from root node to the target with Fibonacci matrices:

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Properties and Relations (3) 

BarningHallTreePosition with "PathWithFibonacciPythagoreanPair" automatically aligns the Pythagorean triples vertically in the node to minimize the width of each node as the triples grows rapidly:

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"PathWithPythagoreanTriple" saves raw data therefore no automatic formatting implemented:

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Use TreeMap and Column or TableForm function to convert the tree into more compact and legible form:

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BarningHallTreePosition properly handles the display of gigantic Pythagorean triples with deep tree levels using the Format function internally. Together with TreeExtract and TreeElementStyle we can create a neat subtree. Specify a Pythagorean triple with huge values:

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Each number is close to 170 digits:

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The path on the tree has 300 levels of depth and would not be a useful visualization:

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Take the last three levels including the target input on the tree:

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Customize the coordinate for the tree nodes to arrange the sibling, parents and grandparent node for the input:

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Possible Issues (3) 

The input must be Pythagorean triple. For instance {5,12,14} is not Pythagorean. The function returns unevaluated:

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The Pythagorean triple must be primitive. For instance {5,12,13} with GCD[5,12,13] = 1. {10,24,26} is Pythagorean but not primitive. The input yields unevaluated form:

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The function uses canonical primitive Pythagorean triple notation as it is compatible with its intermediate triples. Changing the sequence of the input results in an unevaluated result:

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Neat Examples (8) 

Schinzel’s hypothesis H implies that there are infinite number of primitive Pythagorean triples with two primes. This requires that both odd leg a and odd hypotenuse being prime number and the length of the even leg is one less than that of the hypotenuse. Let's take a look at the example:

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Solve for the Pythagorean equation and notice that b is one less than c:

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Actually, given such c, the other two variables are determined in this Diophantine equation:

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For other cases, we choose prime p such that is also a prime number:

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Pick 5 elements and we can check the location of the triples on the tree with obvious pattern:

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For instance, we can visualize the {11,60,61} on the B-H tree:

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The number of 3's in each position for primitive Pythagorean triple with two primes defines an interesting sequence. The sequence is A068501 with each term minus one:

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Plot the curve of the sequence:

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Notice that the tree position of repeating 3's is just a necessary condition for Pythagorean triple to have 2 primes. (7,24,25) is at {3,3} and 25 is not prime. In fact, the tree position of repeating 3's is sufficient and necessary condition for primitive Pythagorean triple with (a,c-1,c) form in our Barning-Hall tree convention. Those triples with two primes are strictly in the subset of the (a,c-1,c) cases.