Re:Why is Pi² so close to 10?
by soVeryTired on Tuesday March 20, 2018 @11:42AM
(#16629534)
Nope. It's one of a few very special numbers though.
Every integer has a unique prime factorisation: that's something kids learn in primary school. The idea of factorisation goes a lot deeper though. Define the gaussian integers as integers of the form a + ib, where a and b are intgers. Unique factorisation holds in the Gaussian integers too, but some of the numbers we recognise as prime are no longer prime. For example, 2 = (1 + i)(1 – i).
It turns out unique factorisation is a rare property of number systems. Consider the set of numbers whose ‘integers’ are of the form a + b sqrt(-5), with a and b standard integers. In this system, 6 = 2x3 = (1 + sqrt(-5)) x (1 - sqrt(-5)). But none of those factors can be factorised further, so 6 has two distinct factorisations.
It turns out there are only nine positive numbers n so that unique factorisation holds in the system a + b*sqrt(-n). 163 is the largest!
Every integer has a unique prime factorisation: that's something kids learn in primary school. The idea of factorisation goes a lot deeper though. Define the gaussian integers as integers of the form a + ib, where a and b are intgers. Unique factorisation holds in the Gaussian integers too, but some of the numbers we recognise as prime are no longer prime. For example, 2 = (1 + i)(1 – i).
It turns out unique factorisation is a rare property of number systems. Consider the set of numbers whose ‘integers’ are of the form a + b sqrt(-5), with a and b standard integers. In this system, 6 = 2x3 = (1 + sqrt(-5)) x (1 - sqrt(-5)). But none of those factors can be factorised further, so 6 has two distinct factorisations.
It turns out there are only nine positive numbers n so that unique factorisation holds in the system a + b*sqrt(-n). 163 is the largest!
https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem