OFFSET
1,2
COMMENTS
Define two parameters E and M for a rational approximation j/k for an irrational number x: E = abs(j/k - x) (the absolute error) and M = 1/(sqrt(5)*k^2). Hurwitz's theorem states that every real number has infinitely many rational approximations that satisfy E/M < 1, making each such approximation a "strong approximation". This sequence lists the denominators of such numbers for the irrational number Pi.
LINKS
Jon E. Schoenfield, Magma program with explanation of algorithm
Wikipedia, Hurwitz's theorem (number theory)
EXAMPLE
22/7 ~ 3.1428571 and E/M ~ 0.1385.
355/113 ~ 3.1415929 and E/M ~ 0.0076.
From Jon E. Schoenfield, Aug 06 2021: (Start)
k j E = |j/k - Pi| M = 1/(sqrt(5)*k^2) E/M
----- ------ -------------- ------------------- -------
1 3 0.141592653590 0.44721359549995794 0.31661
7 22 0.001264489267 0.00912680807142771 0.13855
14 44 0.001264489267 0.00228170201785693 0.55419
113 355 0.000000266764 0.00003502338440755 0.00762
226 710 0.000000266764 0.00000875584610189 0.03047
339 1065 0.000000266764 0.00000389148715639 0.06855
452 1420 0.000000266764 0.00000218896152547 0.12187
565 1775 0.000000266764 0.00000140093537630 0.19042
678 2130 0.000000266764 0.00000097287178910 0.27420
791 2485 0.000000266764 0.00000071476294709 0.37322
904 2840 0.000000266764 0.00000054724038137 0.48747
1017 3195 0.000000266764 0.00000043238746182 0.61696
1130 3550 0.000000266764 0.00000035023384408 0.76167
1243 3905 0.000000266764 0.00000028944945791 0.92163
33215 104348 0.000000000332 0.00000000040536522 0.81810
(End)
MATHEMATICA
a={}; For[k=1, k<=10^6, k++, If[Abs[Round[k Pi]/k-Pi]Sqrt[5] k^2<1, AppendTo[a, k]]]; a (* Stefano Spezia, Aug 07 2021 *)
PROG
(Magma) // See Links.
(PARI) is(k) = my(j=round(Pi*k)); abs(j/k - Pi)*sqrt(5)*k^2 < 1; \\ Jinyuan Wang, Aug 06 2021
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
June Richardson, Jul 22 2021
EXTENSIONS
a(17)-a(19) from Jinyuan Wang, Aug 06 2021
a(20)-a(31) from Jon E. Schoenfield, Aug 06 2021
STATUS
approved