lpetrich
Contributor
Almost Integer -- from Wolfram MathWorld discusses several real numbers that are surprisingly close to integers, and it mentions some particularly odd ones, including one attributed to mathematician Srinivasa Ramanujan, who lived a century ago (Ramanujan Constant -- from Wolfram MathWorld). I calculated them with Mathematica, and I verified that that is indeed the case.
exp(pi*sqrt(22)) = 2 508 951.998 257...
exp(pi*sqrt(37)) = 199 148 647.999 978 047...
exp(pi*sqrt(58)) = 24 591 257 751.999 999 822 213...
exp(pi*sqrt(163)) = 262 537 412 640 768 743.999 999 999 999 250 073...
These results can be derived with the help of "modular functions", in particular, the "j-function" (j-Function -- from Wolfram MathWorld). That last one has the help of the algebraic field Q(sqrt(-163)), the rational numbers extended with sqrt(-163), and the others are similar.
exp(pi*sqrt(22)) = 2 508 951.998 257...
exp(pi*sqrt(37)) = 199 148 647.999 978 047...
exp(pi*sqrt(58)) = 24 591 257 751.999 999 822 213...
exp(pi*sqrt(163)) = 262 537 412 640 768 743.999 999 999 999 250 073...
These results can be derived with the help of "modular functions", in particular, the "j-function" (j-Function -- from Wolfram MathWorld). That last one has the help of the algebraic field Q(sqrt(-163)), the rational numbers extended with sqrt(-163), and the others are similar.