# The Ulam Spiral

Wikipedia: The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963….[1] It is constructed by writing the positive integers in a square spiral and specially marking the prime numbers.

Ulam… emphasized the striking appearance in the spiral of prominent diagonal, horizontal, and vertical lines containing large numbers of primes. …Ulam… noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler’s prime-generating polynomial x2 − x + 41, are believed to produce a high density of prime numbers.[2][3] Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau’s problems. In particular, no quadratic polynomial has ever been proved to generate infinitely many primes, much less to have a high asymptotic density of them, although there is a well-supported conjecture as to what that asymptotic density should be.

In 1932, more than thirty years prior to Ulam’s discovery, the herpetologist Laurence Klauber constructed a triangular, non-spiral array containing vertical and diagonal lines exhibiting a similar concentration of prime numbers. Like Ulam, Klauber noted the connection with prime-generating polynomials, such as Euler’s.[4]