37--36--35--34--33--32--31 | | 38 17--16--15--14--13 30 | | | | 39 18 5-- 4-- 3 12 29 | | | | | | 40 19 6 1-- 2 11 28 | | | | | 41 20 7-- 8-- 9--10 27 | | | 42 21--22--23--24--25--26 | 43--44--45--46--47--48--49...He then circled all of the prime numbers and he got the following picture:
37-- -- -- -- -- --31 | | 17-- -- -- --13 | | | | 5-- -- 3 29 | | | | | | 19 -- 2 11 | | | | | 7-- -- -- 41 | | -- --23-- -- -- | 43-- -- -- --47-- -- ...To his surprise, the circled numbers tended to line up along diagonal lines. The following image illustrates this. This is a 200×200 Ulam spiral, where primes are black. Black diagonal lines are clearly visible.
It appears that there are diagonal lines no matter how many numbers are plotted. This seems to remain true, even if the starting number at the center is much larger than 1. This implies that there are many integer constants b and c such that the function:
generates an unexpectedly-large number of primes as n counts up {1, 2, 3, ...}. This was so significant, that the Ulam spiral appeared on the cover of Scientific American in March 1964.
At sufficient distance from the centre, horizontal and vertical lines are also clearly visible.
External links: