nearly equal arithmetical expressions This is for teaching math. I'm wondering if someone knows some striking near-equalities between simple arithmetic expressions. I vaguely remember that such things exist (e.g., numbers that look alike out 10 digits, but that are really different), but I don't remember where I saw them or what they are. I think there are even some famous historical instances where it was debated whether certain expressions were equal or not.
One possibility I am particularly interested in is integer combinations of (integer) square roots that turn out to be very very close to zero (but that are nonzero). Does anyone know how to construct these? I am assuming they exist because I read somewhere (and forgotten again!) that there is currently no efficient algorithm---or maybe even no algorithm at all, can't remember---for determining the sign of such a sum.
Thank you!
PS: I'm also interested in sources (e.g., book of number puzzles or relevant number theory, etc).
 A: $e^{\sqrt{163}\pi}=262537412640768743.99999999999925$ is almost an integer.  It was conjectured for a long while this number could in fact be an integer.
You can find more on this here:
https://www.youtube.com/watch?v=DRxAVA6gYMM
Why is $e^{\pi \sqrt{163}}$ almost an integer?
A: Imagine the following scenario: A physicist modeling a certain system finds out that the number
$$\alpha:={1\over10}\sum_{n=-\infty}^\infty e^{-(n/10)^2}$$
is relevant for his problem. He numerically computes partial sums $s_N:=\sum_{|n|\leq N}$ and obtains, e.g.,
$$s_{10}=1.529\ldots,\quad s_{20}=1.7659\ldots,\quad s_{40}=1.772454\ldots,\quad s_{100}=1.77245385090552\ldots\ .$$
From the website Inverse Symbolic Calculator he learns that his $s_{100}$ agrees with $\sqrt{\pi}$ in all given decimal places. Is this just a numerical coincidence?
Our scientist then turns to a mathematician and is told that the number $\alpha$ is a value of  the theta function $\theta(x):=\sum_{k=-\infty}^\infty e^{-k^2\pi x}$:
$$\alpha:={1\over10}\>\theta\bigl({1\over 100\pi}\bigr)\ .$$ 
Now comes the upshot:  Jacobi's famous theta transformation allows to rewrite $\alpha$ in the form
$$\alpha={\sqrt{100\pi}\over 10}\theta(100\pi)=\sqrt{\pi}\sum_{k=-\infty}^\infty e^{-100\pi^2k^2}=\sqrt{\pi}(1+\epsilon)\ ,$$
where $\epsilon:=\sum_{k\ne0} e^{-100\pi^2k^2}<10^{-428}$. In other words: The number $\alpha$ resembles $\sqrt{\pi}$ up to more than $400$ decimal places, but is $\ne\sqrt{\pi}$.
A: additional reference: Section 1.4 (High Precision Fraud) in Borwein, Bailey, Girgensohn: Experimentation in Mathematics, A.K.Peters 2004.
