As an HVAC engineer by trade, [Carlos Paris] spends a lot of time in AutoCAD designing all those hidden pipes, tubes, and ducts hidden in a building’s rafters. One day, [Carlos] read of an open contest – the prize was over a million dollars – to generate a prime number with a billion digits. [Carlos] misheard this as, ‘a prime number greater than one billion’ and of course said this was a trivially easy task and opened up his favorite tool – AutoCAD – in an effort to discover the largest prime ever. [Carlos] never generated a remarkably large prime, but he did come up with a very, very cool visualization of prime numbers on a number line, as well as a great justification of the twin prime conjecture, a problem in mathematics that has remained unsolved for several generations.

[Carlos] started his investigations into the properties of prime numbers by drawing a series of circles on a number line in AutoCAD. These circles were of diameters of all the integers, and going down the number line, these circles started to have an interesting, chaotic pattern (see above picture). [Carlos] found that whenever two circles intersected, that position was a prime number. It’s really nothing more than a Sieve of Eratosthenes, but it’s a very cool-looking visualization nonetheless.

Looking deeper into his graph, [Carlos] discovered there were certain primes that had another prime number just two places down the number line. For example, the numbers 3 and 5, 29 and 31, and 41,and 43 are twin primes, as the difference between the primes is only 2. The idea there are infinitely many twin primes is a famous unsolved problem in mathematics – it’s obvious it must be true, but no mathematician has yet come up with a proof of this conjecture.

[Carlos] looked at his number line and simplified it to a generic prime number. By taking a generic number line and overlaying the multiples of other prime numbers on this graph, [Carlos] had a very, very clever way of understanding exactly *how* twin primes come into existence.

In the end, [Carlos] is no closer to proving the twin prime conjecture than anyone else. We’ve got to hand it to him, though, for nerding out with an engineer’s favorite tool – AutoCAD – and managing to derive some fairly obscure mathematics on his own.

After the break you can see [Carlos]’s videos describing the though process that went into his creation. Very, very cool work.

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