## 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 $$10^{-12}.$$