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 n + 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 , 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.
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