pyth. trippel and the calculation of pi with them

[bhelmes]

A peaceful and pleasent night for everyone,

i read a notice of the calculation of pi by the help of pyth. trippel:

Let the number of triples with hypotenuse less than N be denoted A(N)

N/A(N) = 2pi

The result was proven by Lehmer (1900).

Verification by myself:

2^n = 5. N = 11 ~pi = 2.90909090909090917165
2^n = 6. anzahl = 18 ~pi = 3.55555555555555535818
2^n = 7. anzahl = 38 ~pi = 3.36842105263157876038
2^n = 8. anzahl = 81 ~pi = 3.16049382716049365172
2^n = 9. anzahl = 163 ~pi = 3.14110429447852768092
2^n = 10. anzahl = 323 ~pi = 3.17027863777089802255
2^n = 11. anzahl = 653 ~pi = 3.13629402756508435246
2^n = 12. anzahl = 1310 ~pi = 3.12671755725190836372
2^n = 13. anzahl = 2607 ~pi = 3.14230916762562317857
2^n = 14. anzahl = 5211 ~pi = 3.14411821147572423385
2^n = 15. anzahl = 10426 ~pi = 3.14291195089200092738
2^n = 16. anzahl = 20863 ~pi = 3.14125485308920104899
2^n = 17. anzahl = 41728 ~pi = 3.14110429447852768092
2^n = 18. anzahl = 83429 ~pi = 3.14212084526963053577
2^n = 19. anzahl = 166871 ~pi = 3.14187605995050089902
2^n = 20. anzahl = 333787 ~pi = 3.14145248317040515218
2^n = 21. anzahl = 667584 ~pi = 3.14140542613364015523
2^n = 22. anzahl = 1335065 ~pi = 3.14164778493930985093
2^n = 23. anzahl = 2670147 ~pi = 3.14162778303966039317
2^n = 24. anzahl = 5340303 ~pi = 3.14162248846179714690
2^n = 25. anzahl = 10680690 ~pi = 3.14159778066772821248
2^n = 26. anzahl = 21361461 ~pi = 3.14158586812016293877
2^n = 27. anzahl = 42722757 ~pi = 3.14159800127131294545
2^n = 28. anzahl = 85445541 ~pi = 3.14159700855542611819
2^n = 29. anzahl = 170891241 ~pi = 3.14159408556229058362
2^n = 30. anzahl = 341782682 ~pi = 3.14159224720461427438
2^n = 31. anzahl = 683565237 ~pi = 3.14159283088294305486
2^n = 33. anzahl = 2734261194 ~pi = 3.14159254823553624192
2^n = 34. anzahl = 5468521887 ~pi = 3.14159283605332317890
2^n = 35. anzahl = 10937044186 ~pi = 3.14159271770907722043
2^n = 36. anzahl = 21874088616 ~pi = 3.14159268266539415393
2^n = 37. anzahl = 43748178397 ~pi = 3.14159259900578557989
2^n = 38. anzahl = 87496355552 ~pi = 3.14159264360030610064
2^n = 39. anzahl = 174992710606 ~pi = 3.14159265254075359408
2^n = 40. anzahl = 349985420298 ~pi = 3.14159266074514009759

Does anyone knows the proof ?

Greetings from the circle
Bernhard

[Dr Sardonicus]

This is for primitive Pythagorean triples. Lehmer's original paper is Asymptotic Evaluation of certain Totient Sums. The result in question starts at the bottom of page 327 and continues at the top of page 328.

[bhelmes]

Originally Posted by Dr Sardonicus

This is for primitive Pythagorean triples. Lehmer's original paper is Asymptotic Evaluation of certain Totient Sums. The result in question starts at the bottom of page 327 and continues at the top of page 328.

A peaceful and pleasant night for you,

i did not get the whole idea for the proof.

The primitive Pythagorean triples make a surjective relation to the lattice of the complex number with n^2-(mI)^2=1 mod p
where mI should be the irrational part and
where p is a prime.
The complex order is of this subgroup is p+1,

Did i miss the clue ?

Greetings from the complex plane
Bernhard