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Integer structure analysis within the framework of modular rings is used to show that the formation of primitive (or “reduced”) Pythagorean triples depends on certain characteristics with these rings. Only integers in the Class 1̅4 of the modular ring Z4 can produce primitive Pythagorean triples. Of these, a prime produces only one primitive Pythagorean triple, while composites produce the same number of primitive Pythagorean triples as their factors, provided the factors are square-free or are not elements of 3̅4 . Class 1̅4 integers were converted to the equivalent Z6 classes in order to isolate those divisible by 3. The numbers of primitive Pythagorean triples in various ranges were estimated and compared with the elder Lehmer’s estimates. The results provide a neat link between the number of primitive Pythagorean triples and the number of primes in the given interval. It was also shown why the major component of a primitive Pythagorean triple is the only component which cannot have 3 as a factor.
- Primitive Pythagorean triples
- Modular rings
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Cite this paper
Leyendekkers, J., & Shannon, A.(2012). The number of primitive Pythagorean triples in a given interval, Notes on Number Theory and Discrete Mathematics, 18(1), 49-57.