Journey into PI 
Approximations using \(\pi\)
An equally interesting problem and diametrically opposed to the previous one, is to obtain integers or rational fractions from expressions involving \(\pi\).
The best known example is probably the expressions of Roy Williams\[e^{\pi\sqrt{n}},n\in\mathbb{N}\]
For some values of \(n\)*, the result of the expression approximates an integer.
| \(n\) | \(e^{\pi\sqrt{n}},n\in\mathbb{N}\) |
| \(1\) | \(23.\, 140\, 692\, 632\, 779\, 269\, 005\) |
| \(2\) | \(85.\, 019\, 695\, 223 \,207 \,217 \,582\) |
| \(3\) | \(230. \,764 \,588 \,319 \,145 \,879 \,240\) |
| \(7\) | \(4071. \,932 \,095 \,225 \,261 \,098 \,524\) |
| \(11\) | \(33506. \,143 \,065 \,592 \,438 \,766 \,681\) |
| \(19\) | \(885479. \,777 \,680 \,154 \,319 \,497 \,537\) |
| \(25\) | \(6635623. \,999 \,341 \,134 \,233 \,266 264\) |
| \(37\) | \(199148647.\, 999 \,978 \,046 \,551 \,856 \,766\) |
| \(43\) | \(884736743. \,999 \,777 \,466 \,034 \,906 \,661\) |
| \(58\) | \(24591257751. \,999 \,999 \,822 \,213 \,241 \,469\) |
| \(67\) | \(147197952743. \,999 \,998 \,662 \,454 \,224 \,506\) |
| \(74\) | \(545518122089. \,999 \,174 \,678 \,853 \,549 \,856\) |
| \(148\) | \(39660184000219160. \,000 \,966 \,674 \,358 \,575 \,246\) |
| \(163\) | \(262537412640768743. \,999 \,999 \,999 \,999 \,250 \,072\) |
| \(232\) | \(604729957825300084759. \,999 \,992 \,171 \,526 856\, 430\) |
| \(268\) | \(21667237292024856735768. \,000 \,292 \,038 \,842 \,412 \,959\) |
| \(522\) | \(14871070263238043663567627879007. \,999 \,848 \,726 \,482 \,794 \,814\) |
| \(652\) | \(68925893036109279891085639286943768. \,000 \,000 \,000 \,163 \,738 \,644\) |
| \(719\) | \(3842614373539548891490294277805829192. \,999 \,987 \,249 \,566 \,012 \,187\) |
* In particular the Heegner numbers \(\left\{ 1,2,3,4,11,19,43,67,163\right\}.\)
Noteworthy is the value \(e^{\pi\sqrt{163}}\) that approximates an integer with an error less than \(10^{-12}.\)
