Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Question

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they would appear unequal?

Of course, I don't mean such things like $\arccos(-1+10^{-10000})\approx\pi$, which are clearly ad hoc and uninteresting. I mean something more like $\pi^4+\pi^5\approx e^6$ (which holds with relative error $\approx 4\times10^{-8}$), but would like to have some similarly simple approximate equality valid with relative error of $10^{-100}$ or even smaller.

@Peter googled for "unexpected approximate equalities", opened wikipedia link — Ruslan, Feb 14 '14 at 10:21
$3487^{12}+4365^{12} \approx 4472^{12}$, and this is from Homer Simpson — Math137, Feb 14 '14 at 10:22
@math137 These numbers differ even in most significant digit... — Ruslan, Feb 14 '14 at 10:25
@math137 still $1.56\times10^{-44}$ vs $1.96\times 10^{-44}$. Not too close. — Ruslan, Feb 14 '14 at 10:27
@math137 To reveal Homer's genius rather compute $\sqrt[12]{3487^{12}+4365^{12}}\approx 4472.000000007$ — Hagen von Eitzen, Feb 14 '14 at 10:37
@hagen but this is not "unexpected". There is a theory explaining why this number is nearly an integer. — Peter, Feb 14 '14 at 10:39
PARI gives the following result : ? (3487^12+4365^12)^(1/12) %6 = 4388.841178878421569355007603 ? Is there a typo somewhere ? — Peter, Feb 14 '14 at 10:42
? (3987^12+4365^12)^(1/12) %8 = 4472.000000007059290738213530 ? So, 3487 must be replaced by 3987. — Peter, Feb 14 '14 at 10:45
? 3987^12+4365^12 %9 = 63976656349698612616236230953154487896987106 ? 4472^12 %10 = 63976656348486725806862358322168575784124416 ? Shows that a normal table calculator cannot distinguish these numbers. Related to the "near misses" concerning fermats equation $x^n + y^n = z^n$ — Peter, Feb 14 '14 at 10:47
A very nice example is also this: http://en.wikipedia.org/wiki/Borwein_integral — Denis, Feb 14 '14 at 10:52
You can find several examples in Chapter 10, titled "Danger! Always Exercise Caution When Using the Computer" in "The Computer As Crucible: An Introduction To Experimental Mathematics" by Jonathan Borwein and Keith Devlin. — awkward, Feb 15 '14 at 01:38
A 42 billion digits coincidence: http://oldweb.cecm.sfu.ca/~pborwein/SLIDES/COGNOS.pdf, p. 25. — Martín-Blas Pérez Pinilla, Feb 17 '14 at 07:48
@Martín-BlasPérezPinilla It and its description is given in Borwein, Borwein "Strange Series and High Precision Fraud". But it's obviously no coincidence: $10^5$ and its square inside exponential is too similar to that $10^{-10000}$ in the example with $\arccos$ I gave in the question. If this number is enlarged (e.g. to $10^6$), we'd get even higher precision of coincidence (namely, 4.2 trillion digits). — Ruslan, Feb 17 '14 at 08:38

score 1 · Answer 1 · edited Aug 18 '20 at 19:32

1

$e^{\pi\sqrt{163}}\approx 262\,537\,412\,640\,768\,744$ to 12 decimal places.

edited Aug 18 '20 at 19:32

Ruslan

6,985

answered Aug 18 '20 at 19:26

Erich Friedman

507
2
3

This doesn't satisfy the requirement of relative error $\le10^{-100}$ though... – Ruslan Aug 18 '20 at 19:37

Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

1 Answers1