Ramanujan found the exact value of the Ramanujan G-function G69=(5+√23√2)1/12(3√3+√232)1/8(√2+3√34+√6+3√34)1/2Note the fundamental units Un U23=24+5√23=(5+√23√2)2U69=25+3√692=(3√3+√232)2 and how he uses the squared version. As a second example
G77=(8+3√7)1/8(√7+√112)1/8(√2+√114+√6+√114)1/2
and fundamental units U7=8+3√7=(3+√7√2)2U77=9+√772=(√7+√112)2
Ramanujan mostly uses the squared version of the Un to get "simpler" expressions with smaller integers. For prime p≡3mod4, one can always do since
x2−py2=−2x2−py2=+2 are solvable by p≡3mod8 and p≡7mod8, respectively. Checking U67 and U163, yields the reductions U67=48842+5967√67=(221+27√67√2)2U163=64080026+5019135√163=(8005+627√163√2)2And from eπ√67≈52803+744 and eπ√163≈6403203+744, we find the relations 5280=24(221−1)640320=80(8005−1) though it may be just coincidence.
No comments:
Post a Comment