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Some toots about number 163 

I've noticed that I'm close to \(163)\. toot, so I'll toot something very cool about this number.
\(163)\ is largest of nine Heegner numbers, and I'll explain what those are as I understand it. Gauss :gauss: wanted to identify all perfect Pythagorian :pythagoras: triplets which are whole numbers \(a, b ,c\) that satisfy \(a²+b² = c²\). To do that he had to involve complex numbers and come up with new factorisation system.

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Some toots about number 163 

So factorised \(a²+b² = c²\) in new system (which involves imaginary numbers) is now \(a²+b²=(a-ib)+(a+ib)\) and that system had to have same property of unique factorisation of whole numbers, unique factorisation means that any whole number \(w\) can be written as unique product of prime numbers, and this works for \(\sqrt{-1}\)

Some toots about number 163 

Heegner numbers are numbers which don't show unique factorisation in that new defined system, so for example \(6=2*3\) is also \(6=(1+\sqrt{-5})(1-\sqrt{5})\), here 5 is Heegner number, and \(163\) also has that property and its last proven number to have that property.

Some toots about number 163 

Also \(163\) has property of giving almost whole number when used in \(e^{\sqrt{163}\pi}=262537412640768743.9999999999992500 ...\) as one of Ramanujan constants. It has some more properties as being one of "lucky" and "fortunate" math numbers, and gives good aproximations of \(e\) and \(\pi\) . Pretty stacked up number, if you ask me.

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