Dyson, one of the most highly-regarded scientists of his time, poignantly informed the young man that his findings into the distribution of prime numbers corresponded with the spacing and distribution of energy levels of a higher-ordered quantum state. Mathematics Problem That Remains Elusive —And Beautiful By Raymond Petersen
Robert Sacks devised the Sacks spiral, a variant of the Ulam spiral, in 1994. It differs from Ulam’s in three ways: it places points on an Archimedean spiral rather than the square spiral used by Ulam, it places zero in the center of the spiral, and it makes a full rotation for each perfect square while the Ulam spiral places two squares per rotation. Certain curves originating from the origin appear to be unusually dense in prime numbers; one such curve, for instance, contains the numbers of the form n2 + n + 41, a famous prime-rich polynomial discovered by Leonhard Euler in 1774. The extent to which the number spiral’s curves are predictive of large primes and composites remains unknown.
A closely related spiral, described by Hahn (2008), places each integer at a distance from the origin equal to its square root, at a unit distance from the previous integer. It also approximates an Archimedean spiral, but it makes less than one rotation for every three squares.
It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.). See more info on Mersenne Prime.