Why do prime numbers make these spirals? (2019)
One of the thrills of studying the primes is the discovery that all primes greater than 3 are of the form 6k+1 or 6k-1. And for primes greater than 2, all primes are of the form 4k+1 or 4k-1. It is something that is commonly rediscovered by students, and that new independent finding was quite exciting for me.<p>The reasoning, which is in the article here, is that you can make any whole number you wish if the number is of the form 6k+1, 6k-1, 6k+2, 6k-2, 6k+3, or 6k-3. But you cannot make primes with the numbers of the form 6k+2, 6k-2 (they would always have to be divisible by 2), and you cannot makes primes with numbers of the form 6k+3, 6k-3 because they are always divisible by 3. So what are you left with? All primes >3 must of the form 6k+1 or 6k-1. And that factor 6 is just a bit less than 2 pi (a full turn in radians) so you get spirals from the offset. They are also a pixel or two off, but that is imperceptible.<p>The same logic is a nice exercise to apply to the problem of why primes >2 can only be of the form 4k+1 or 4k-1. Apply the same logic as above.