MacTutor
Ulam's spiral
In 1963, during a boring lecture, Stan Ulam was doodling on his squared notepad. He arranged the positive integers in a square spiral pattern and noticed that when he marked the prime ones they seemed to concentrate themselves in diagonal lines.
He investigated it further using the MANIAC computer and found quite distinct lines of primes. In fact the numbers on the diagonal, horizontal or vertical lines are all of the form for integers and quadratic formulae like this are known to sometimes produce high densities of primes.
For example on this diagram:
the primes of the form have been marked in blue and most of them are on a distinct diagonal straight line.
Shortly afterwards Martin Gardner used the spiral in his Scientific American column and the picture appeared on the cover.
He investigated it further using the MANIAC computer and found quite distinct lines of primes. In fact the numbers on the diagonal, horizontal or vertical lines are all of the form for integers and quadratic formulae like this are known to sometimes produce high densities of primes.
For example on this diagram:
the primes of the form have been marked in blue and most of them are on a distinct diagonal straight line.
Shortly afterwards Martin Gardner used the spiral in his Scientific American column and the picture appeared on the cover.
Last Updated April 2024