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See A198453.

The definition can be generalized to define Pythagorean k-triples a<=bA198453 for more about Pythagorean k-triples.

The definition can be generalized to define Pythagorean k-triples a<=bA198453 for more about Pythagorean k-triples.

Refer to Cairo tiling by Stick Cross Method (see details in the links). Each pentagon has four sides of equal length and one side which is either shorter or longer. All sides can be taken to have integral lengths related to primitive Pythagorean triples A103606.

If Pythagorean triple = (a, b, c), the 5-tuple is (s1, s2, s3, s4, s5) with s1 = s2 = s3 = s4 = c and s5 = 2*(b-a). See illustration in the links.

The Shiraishi theorem demonstrated that there were an infinite number of cubic number triples whose sum equaled a cubic number (A226903). Through an analysis of the cubic number triples found by Russell and Gwyther, another way to prove that there are an infinite number of cubic number triples who sum equals a cubic number appeared. For any triple, one can add a zero to the end of the three numbers. The new three numbers will also equal a cubic number. For example, 3^3+4^3+5^3=6^3 can be transformed into 30^3+40^3+50^3=60^3. The number of zeros that are consistently applied to each of the numbers who cubic numbers will always create new cubic numbers. For example, 30^3+40^3+50^3=60^3 can become 300^3+400^3+500^3=600^3 and 3000^3+4000^3+5000^3=6000^3, and so on. Through experiments, the formula holds true for Pythagorean triples and Pythagorean quadruples as well. To apply the method to Pythagorean triples, 3^2+4^2=5^2 can be transformed into 30^2+40^2=50^2, 300^2+400^2=500^2, and so on. For Pythagorean quadruples, 3^2+4^2+12^2=13^2 can be transformed into 30^2+40^2+120^2=130^2 and then to 300^2+400^2+1200^2=1300^2. The property holds even beyond the second and third powers. For example, 3530^4=300^4+1200^4+2720^4+3150^4 just as 353^4=30^4+120^4+272^4+315^4. Additionally, 1440^5=270^5+840^5+1100^5+1330^5 just as 144^5=27^5+84^5+110^5+133^5. Once one set is found, it appears there can be an infinite number of similar sets for any power through this method.

The list of Russell and Gwyther also reveals that the cube of 38 can be represented as the sum of the cubes of nine unique positive integers. This is because 38^3=3^3+4^3+5^3+7^3+14^3+17^3+18^3+24^3+30^3.

The definition can be generalized to define Pythagorean k-triples a<=bA198453 for more about Pythagorean k-triples.

The definition can be generalized to define Pythagorean k-triples a<=bA198453 for more about Pythagorean k-triples.

Procedure: The sequence lists two indices, j and k. Let (a, b, and c) represent the two legs and the hypotenuse of a right-angled triangle. The required computations are as follows:

(1) compute 2*j^2 (call this p, it is c-a)

(2) compute (2*k-1)^2 (call this q, it is c-b)

(3) compute 2j*(2k-1) (call this r, it is a+b-c)

So c = p+q+r; a = c-p; b = c-q.

Most, but not all, of the Pythagorean triples generated here are primitive (i.e., they have sides that are relatively prime). In the first 105 Pythagorean triples, there are 14 that are not primitive. The first exception is the 9th in the list: 27,36,45, where j=3 and k=2.