Timeline for Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?
Current License: CC BY-SA 3.0
6 events
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| Jul 19, 2016 at 20:24 | comment | added | Yaakov Baruch | Also, the seeming uniform distribution of primitive triples would seem, euristically, to imply an asymptotic of $\frac{1}{2 \pi} \int_1^N \lfloor \frac{N}{x}\rfloor dx \simeq \frac{1}{2 \pi} N\log(N)$ for all triples. Or am I wrong? | |
| Jul 19, 2016 at 18:12 | comment | added | Greg Martin | Your parametrization of Pythagorean triples is inaccurate in the details: it requires $u$ and $v$ to be relatively prime and of opposite parity, and then the formula parametrizes all primitive Pythagorean triples, not all Pythagorean triples. In other words, as stated, your method double-counts some triples and omits other triples. | |
| Jul 19, 2016 at 17:30 | comment | added | Benjamin Dickman | Yes, the equivalence to "all Pythagorean triples with bounds on the hypotenuse" makes it straightforward to search for (I posted a few example references after using such a query) ... But I'm [somewhat] wary about the reference you provide as it exists only on vixra ... | |
| Jul 19, 2016 at 11:20 | comment | added | Manfred Weis | on the wolfram page the asymptotic formula $\sum_{i=1}^{n}r(i) = \pi n + O(\sqrt{n})$ is given | |
| Jul 19, 2016 at 7:03 | comment | added | Yaakov Baruch | is there a simple asymptotic for the latter number? | |
| Jul 19, 2016 at 5:23 | history | answered | Manfred Weis | CC BY-SA 3.0 |